**5. Conclusion**

Multivariate statistical analysis techniques were used to establish the relationships between plant diversity, Topography and soil factors. Plant community, structure and biodiversity have been shown to have a high degree of spatial variability that is controlled by both abiotic and biotic factors (Fu et al, 2004).

CCA is the constrained form of CA, and therefore is preferred for most ecological data sets (since unimodality is common). CCA also is appropriate under a linear model, as long as one is interested in species composition rather than absolute abundances (ter Braak and Šmilauer 1998). Correspondence analysis (CA) and canonical correspondence analysis (CCA) are widely used to obtain unconstrained unconstrained or constrained ordinations of species abundance data tables and the corresponding biplots or triplots which are extremely useful for ecological interpretation CA provided a good approximation for species with unimodal distributions along a single environmental gradient. There is a problem with this metric, however: a difference between abundance values for a common species contributes less to the distance than the same difference for a rare species, so that rare species may have an unduly large influence on the analysis (Greig-Smith 1983; ter Braak and Smilauer 1998; Legendre and Legendre 1998).

The most other general ordination technique, nonmetric multidimensional scaling (NMDS), which is based on the rankings of distances between points (Shepard, 1962), circumvents the linearity assumption of metric ordination methods. This method, used in ecological investigations (Kenkel and Orloci, 1986), Comparative studies of ordination techniques have, moreover, demonstrated the superiority of NMDS, and some authors have re commended its use, notwithstanding the computational burden.

The NMDS approach can in fact be tested each time measures of re semblance or dissimilarity are used to classify OTUs, whatever the causes and origins of arrangements found (Guiller et al, 1998).

248 Multivariate Analysis in Management, Engineering and the Sciences

ultimately be governed by this philosophical distinction.

**5. Conclusion**

abiotic and biotic factors (Fu et al, 2004).

Legendre and Legendre 1998).

– and for such a system, NMDS would be a better *a priori* choice. As De'ath (1999) points out, there are two classes of ordination methods - 'species composition restoration' (e.g. NMDS) and 'gradient analysis' (e.g. DCA). The choice between the methods should

Non-metric multidimensional scaling (NMS) (PC-ORD v. 4.25, 1999) was used to identify environmental variables correlated with plant species composition. A random starting location and Sorensen's distance measurement were used with the NMS autopilot slow and thorough method. Stepwise multiple linear regression (S-PLUS, 2000) was used to select models correlating vegetation cover and structure with environmental factors. Environmental explanatory factors that were not significant contributors (as determined from using stepwise

A Monte Carlo test of 30 runs with randomized data indicated the minimum stress of the 2 axes NMS ordination were lower than would be expected by chance ( p = 0.0968). The final stress and instability of the 2-D solution were 23.71 and 0.00001, respectively. The first ordination axis (NMS1) captured 41.9% of the variability in the dataset and the second (NMS2)

Multivariate statistical analysis techniques were used to establish the relationships between plant diversity, Topography and soil factors. Plant community, structure and biodiversity have been shown to have a high degree of spatial variability that is controlled by both

CCA is the constrained form of CA, and therefore is preferred for most ecological data sets (since unimodality is common). CCA also is appropriate under a linear model, as long as one is interested in species composition rather than absolute abundances (ter Braak and Šmilauer 1998). Correspondence analysis (CA) and canonical correspondence analysis (CCA) are widely used to obtain unconstrained unconstrained or constrained ordinations of species abundance data tables and the corresponding biplots or triplots which are extremely useful for ecological interpretation CA provided a good approximation for species with unimodal distributions along a single environmental gradient. There is a problem with this metric, however: a difference between abundance values for a common species contributes less to the distance than the same difference for a rare species, so that rare species may have an unduly large influence on the analysis (Greig-Smith 1983; ter Braak and Smilauer 1998;

The most other general ordination technique, nonmetric multidimensional scaling (NMDS), which is based on the rankings of distances between points (Shepard, 1962), circumvents the linearity assumption of metric ordination methods. This method, used in ecological investigations (Kenkel and Orloci, 1986), Comparative studies of ordination techniques have, moreover, demonstrated the superiority of NMDS, and some authors have re

commended its use, notwithstanding the computational burden.

captured 31.8%, leading a cumulative 73.7% of variance in dataset explained (Fig.11).

selection at α = 0.05) were excluded from the final model (Davies et al, 2007).

In the biplots, where only the first two axes were used, all methods based upon PCA gave a fair representation of the relative numerical importance of the rare species. The weights in CCA are given by a diagonal matrix containing the square roots of the row sums of the species data table. This means that a site where many individuals have been observed contributes more to the regression than a site with few individuals. CCA should only be used when the sites have approximately the same number of individuals, or when one explicitly wants to give high weight to the richest sites. This problem of CCA was one of our incentives for looking for alternative methods for canonical ordination of community composition data.

For the analysis of sites representing short gradients, PCA may be suitable. For longer gradients, many species are replaced by others along the gradient and this generates many zeros in the species data table. Community ecologists have repeatedly argued that the Euclidean distance (and thus PCA) is inappropriate for raw species abundance data involving null abundances (e.g. Orlóci 1978; Wolda 1981; Legendre and Legendre 1998). For that reason, CCA is often the method favoured by researchers who are analysing compositional data, despite the problem posed by rare species.

De-trended correspondence analysis (DCA) is perhaps the most widely used method of indirect vegetation ordination. But direct ordination of vegetation and environment is achieved with canonical correspondence analysis (CCA). CCA is a relatively new method in which the axes of a vegetative ordination are restricted to linear groups of environmental variables (Zhang et al, 2006)

DCA and CA analyses should be run with the 'downweight rare species' option selected. We generally do not recommend NMS with the Euclidean distance measure; it performed the worst empirically, and has no advantages over the other methods (Culman et al, 2008)

Among the widely used ordination techniques for the plant community analysis Canonical Correspondence (CA) has shown to be superior to others such as PCA (Gauch, 1982). Most community data sets are heterogeneous and contain one or more gradients with lengths of at least two or three half-changes, which makes CA results ordinarily superior to PCA results. However, with relatively homogenous data sets with short gradients, PCA maybe better (Palmer, 1993). Despite the considerable superiority of the CA over PCA, CA is not superior to DCA, which corrects its two major faults such as "arch effect" and "compression of end of first axis" (Gauch, 1982; Kent & Coker, 1992).

For complex and heterogeneous data sets, DCA is distinctive in its effectiveness androbustness (Gauch, 1982). Comparative tests of different indirect ordination techniques have shown that DCA provides a good result (Cazzier & Penny, 2002). This study found that DCA provides better results than CA results (Malik & Husein, 2006).

For example all ordination techniques, used in North East rangeland of Semnan, clearly indicated that gypsum, EC, slope are the most important factors for the distribution of the vegetation pattern.

In the present study, combination of CCA, DCA and RA results showed that *Ar.aucheri-As.spp-Br.to, Artemisia sieberi-Erotia ceratoides, Ar.sieberi-Zy. eurypterum and Zy. eurypterum -Ar. sieberi*  types correlated with A.W2, gr2, O.M2 and clay1 factors and clay in 0-20 depth indicates *Ar.aucheri-As.spp-Br.to* type. *H.strobilaceum* type has strong relationship with soil salinity and heavy texture. This species showed a trend to high soluble rate, salinity and clay percent. S. rosmarinus types indicate soils with light texture and this type directly related to pH and lime percentage while St.barbata-A.aucheri type shows an inverse relation with these factors.

I fact, analysis with DCA gave results similar to CCA, suggesting that there is a relatively strong correspondence between vegetation and environmental factors; with the difference that the DCA is less isolated the site. CCA better shows differences between types. RA shows relationship between sites and factors, like the CCA analysis. RA axis 1 has an eigenvalue of 0.86. RA axis 2 with an eigenvalue of 0.017 is less important. Total variance (inertia) in the species data is 0.8887.In this method eigenvalue of RA axis1 was higher than CCA and DCA axis1. This study reflects that a spatial approach dealing with the most distinctive species of vegetation communities can yield similar results to those obtained with costly physico-chemical analysis and based on complex matrices of plant communities.

Similarity as this study, also Jafari et al (2003) in their study in Hoz-e-Soltan Reigion of Qom Province, showed that PCA analysis indicates that Halocnemum strobilaceum type has direct relationship with Salinity, Lime, pH and Loam.

May this series of papers serve to enhance the understanding and the proper and creative use of ordination methods in community ecology. Finally, understanding relationships between environmental variables and vegetation distribution in each area helps us to apply these findings in management, reclamation, and development of arid and semi-arid grassland ecosystems (Alisauskas, 1998). The ability to factor out covariables and to test for statistical significance further extends the utility of CCA.

Understanding the relationships between ecological variables and distribution of plant communities can provide guidance to sustainable management, reclamation and development of this and similar regions. In this sense, these results increase our understanding of distribution patterns of desert vegetation and related major environmental factors in the North East of Semnan. The results will also provide a theoretical base for the restoration of degenerated vegetation in this area. Understanding the indicator of environmental factors of a given site leads us to recommend adaptable species for reclamation and improvement of that site and similar sites (Zhang et al, 2005)


*3.4. Raman Chemical Mapping (Band Ratios)*

The intensity ratios of specific Raman bands related to the oil, styrene, imide, and cellulose components are further detailed to provide information on the local chemical composition and distribution of these elements over the paper coating.

The distribution of oil is evaluated from the intensity ratios of Raman bands for oil (1655 cm−1), relative to styrene (1602 cm−1) and cellulose (1095 cm−1), as illustrated in Figure 6: the styrene band is chosen as a reference band for the distribution of oil within the coating, while the cellulose band is chosen as a reference for the distribution of coating over the paper substrate. The progressive increase in oil at the surface for different SMI/oil coatings agrees with previous calculations (see Table 2). The band intensity ratios and surface coverage of oil is comparable for thin and thick coatings when expressing the relative intensity of oil versus styrene within the coating itself: indeed, the same coating dispersions for deposition of thin and thick coatings were used. Contrarily, the differences in oil exposure between thin and thick coatings become clear when using the relative band intensity of oil versus cellulose. The surface patterns and locations covered with oil in the coating and those covered with oil are relative to the paper substrate overlap. Interestingly, the inhomogeneities for thin coatings observed in the average intensity maps are not reflected in the oil distribution over the surface. It is obvious that the inhomogeneities of the coating relative to the paper substrate are not expressed when taking the coating (styrene) as a reference. However, homogeneous distribution of the oil is also observed when taking the substrate (cellulose) as a reference. Therefore, the oil seems to be homogeneously spread over the surface and provides a continuous phase. The coating defects in thin coatings with striations can likely be attributed to the viscosity of the coating dispersions, where the presence of a free oil phase locally enhances the formation of a continuous coating due to better flow properties.

The distribution of organic coating moeities is evaluated from the intensity ratios of Raman bands for imide (1765 cm−1), relative to styrene (1602 cm−1) and cellulose (1095 cm−1), as represented in Figure 7. The relative intensity of imide progressively decreases for different SMI/oil coatings, in agreement with previous calculations (see Table 2). The relative intensity of imide is comparable for thick and thin coatings when expressed against styrene, while the differences in imide between thick and thin coatings become clear relative to the paper substrate: the thick coatings are relatively homogeneous with even distribution of the imide, while thin coatings show striations at the same locations as observed in the average intensity maps. Therefore, the distribution of the organic imide phase plays a major role in the coating homogeneity. The imide phase is likely more rigid and viscous compared to the oil phase, with more risk of coating defects. There is good agreement in the surface maps of oil/cellulose versus imide/cellulose for the thin coatings, confirming that an amount of free oil is present between the imide phase and contributes to the formation of a homogeneous coating layer by 'filling' the non-imide zones as a binder. The complementarity of the oil and imide phases is visible for thin coatings with poly- and monounsaturated oils (e.g., SMI/CO, SMI/RO), where the calculated amounts of free oil are relatively small. The latter is less visible for thin coatings with polyunsaturated oils (e.g., SMI/SO) as there is only a limited amount of free oil, while it is partly visible for the coatings with unsaturated oil (e.g., SMI/HCO) as the oil coverage is much larger than the zones in between the imide phase.

**Figure 6.** Raman maps (5 × 5 mm2) with band intensity ratios illustrating exposure of oil at the surface for SMI/oil paper coatings (left two columns: oil/styrene; right two columns: oil/cellulose).

**Figure 7.** Raman maps (5 × 5 mm2) with band intensity ratios illustrating imide moieties at the surface for SMI/oil paper coatings (left two columns: imide/styrene, right two columns: imide/cellulose).

The distribution of organic coating moieties from the intensity ratios of Raman bands for styrene (1602 cm−1 and 1000 cm−1), relative to cellulose (1095 cm−1), are shown in Figure 8. The maximum intensity of styrene bands is comparable for all SMI/oil coating types and does not strongly differ with the oil type, because all copolymers contain a fixed amount of 74 mol % styrene that remains inert during the synthesis of the SMI/oil nanoparticles. The differences in intensities of styrene therefore mainly correspond to variations in coating coverage. In particular, the effects of thin and thick coatings are visualized. The locations for coverage of styrene components represented by Raman bands at 1602 cm−1 and 1000 cm−1 are in agreement, but slight differences in intensity may be attributed to the specific orientation of the styrene parts. The styrene-related absorption bands are characteristic for either the aromatic C=C stretch (1602 cm−1) or the aromatic C–H bending (1000 cm−1). Their intensities depend on the bonding length of the π-conjugation between the vinyl group and benzene ring and change with local conformation of the aromatic styrene groups [25]. The styrene orientation in the SMI copolymer is induced by a specific molecular structure of the high-molecular-weight SMA precursor and induces nanoparticle formation through self-assembly [26]. Therefore, the variations in ratio of both styrene bands at 1602 and 1000 cm−1 can eventually be attributed to configuration or orientation effects: the influences of segment orientation in styrene bloc copolymers with polarized Raman scattering were detected [27]. In particular, the locations corresponding to the aromatic parts (1602 cm−1) are more confined that the aliphatic parts of the styrene, as the aromatic styrene groups are likely more influenced by self-organization within the nanoparticles.

**Figure 8.** Raman maps (5 × 5 mm2) with band intensity ratios illustrating styrene moieties at the surface for SMI/oil paper coatings (left two columns: 1602 cm−1/cellulose, right two columns: 1000 cm−1/cellulose).

In conclusion, the locations for styrene coverage strongly overlap with the previous locations for imide coverage (see Figure 6). The chemical mapping for the band ratios of organic coating moieties (imide and styrene) generally corresponds with the chemical mapping based on average intensities: the areas with high intensity in styrene (see Figure 7) overlap with the areas of high average intensity (see Figure 4). In case of coatings with polyunsaturated oils (SMI/SO, SMI/CO, SMI/SfO), the surface coverage with high intensities of band ratios for oil moieties (see Figure 5) overlap with the areas of styrene coverage (see Figure 7), as the oil is largely encapsulated in the organic parts. For coatings with monounsaturated oils (e.g., SMI/CaO), the surface areas with high intensities of oil moieties (see Figure 5) are complementary to the surface areas with high intensities of styrene moieties (see Figure 7), as there is a large amount of free oil in between the organic coating phase.

### *3.5. Raman Chemical Mapping (Single Bands)*

The surface maps with intensities of single Raman bands provide direct chemical information on the surface composition of coated papers. As only a single band is considered, the spectra are first normalized (cellulose region 1170–1050 cm−1) and baseline-corrected before selecting a specific band. An example of chemical surface maps for different single Raman bands of thick and thin coatings of SMI/SO and SMI/HCO is shown in Figure 9 (the maps for other coating types are given in Figures S2–S5 in Supplementary Materials). The selected Raman bands provide complementary information on the surface coverage of different chemical components.

The surface areas covered with high intensities of the styrene band (1000 cm−1) are similar to those covered with high intensities of the imide band (1765 cm−1). The same surface areas have also been covered by high intensities in previous surface maps with band intensity ratios of organic coating moieties. The surface maps of the cellulose bands (1378 cm−1 and 1095 cm−1) are complementary to the holes observed in the surface maps with organic coating deposits, indicating that the paper is locally penetrating through the coating at those locations. Both cellulose bands cover the same surface areas and exposure of cellulose bands at the surface is more intense for thin coatings than for thick coatings. The intensity of the single band related to hydroxyl groups (3100 cm−1) is highest at locations where cellulose fibers are exposed and lowest at locations covered by the coating. The exposure of cellulose hydroxyl groups at the surface is evidently more intense for thin coatings than for thick coatings. Therefore, interactions between the coating and the paper substrate are expected to happen through hydrogen bonding between hydroxyl groups at the cellulose fibers and the imidized SMI/oil nanoparticles. In a search for other complementary bands, the locations with low intensities in the 1765 cm−1 band (imide) correspond to locations with high intensities of the 1565 cm−1 band. The latter band can be assigned to the amide II band (NH bend + CN stretch) and represents amic acid groups as a remaining intermediate product after imidization (i.e., ammonolyzed maleic anhydride). The non-imidized parts obviously induce minor local defect spots and influence the formation of a fully homogeneous coating layer. These parts do not interact well with the paper substrate as they appear at similar surface areas with high intensities of hydroxyl bands. The surface areas with high intensities of the single oil band (1655 cm−1) fully overlap with the imide zones for coatings with polyunsaturated oil (e.g., SMI/SO), corresponding to previous conclusions that oil is embedded in the organic phase and there is very little free oil. The overlap of oil coverage and imide locations is much less for coatings with monounsaturated oil (see Figures S3–5 in Supplementary Materials). In particular, the oil coverage largely exceeds the locations with imide deposits in the case of unsaturated oil types (e.g., SMI/HCO) due to the large amount of free oil.

**Figure 9.** Raman maps (5 × 5 mm2) with single wavenumbers representing different coating and substrate components at the surface for SMI/oil paper coatings.

The surface maps with simultaneous coverage of the different coating components have been constructed after image processing (Figure 10). The summation images with imide and oil coverage ("imide ADD oil") illustrate the degree of overlap and inclusion of oil within the organic coating phase, with following color code (see online for color figures: red = oil, black = imide). The quantification of thick and thin coatings is visible from the surface areas covered with the red phase. Almost full inclusion of the red surface areas within the black surface areas is recognized for SMI/SO, SMI/MO, and SMI/RO coatings, while the amount of red surface areas outside the black surface areas increases for SMI/SfO, SMI/CaO, and SMI/HCO coatings, which is in parallel with the localization of free oil. The difference in images between oil and imide ("oil SUBTRACT imide") directly illustrate the surface locations with free oil, following the color code (see online for color figures: blue = free oil). From the blue surface areas it can be observed that the amount of free oil increases in the formerly reported order of SMI/oil coating types, and is higher for thick coatings than for thin coatings. In conclusion, the processed images and original Raman surface maps allow us to accurately visualize the different components and local chemical composition at the coated paper surface, and systematically illustrate the effect of oil encapsulation depending on the degree of saturation of the vegetable oil.

**Figure 10.** Processed images of single Raman wavenumbers for imide (1765 cm−1) and oil (1655 cm−1), illustrating lateral distribution of oil within the organic phase (+) and free oil (−).

#### **4. Conclusions**

Dispersive Raman spectroscopy and microscopy provide an efficient analytical tool for the characterization, discrimination, and quantification of thick and thin barrier coatings on paper with nanoparticle deposits of poly(styrene-co-maleimide) that are filled with six different types of vegetable oils, soy oil, corn oil, rapeseed oil, sunflower oil, castor oil, and hydrogenated castor oil. Depending on the degree of oil saturation (iodine value), the imide content gradually decreases and the amount of free oil gradually increases.

A statistical classification of the coatings can be made by the partial least squares model with three principal components (PC), as each of them include information about the different coating components, i.e., oil phase (PC-1), cellulose substrate (PC-2), and the organic coating components (PC-3). The overall classification of the paper coatings in score plots is based on the degree of saturation for the different oils and coating thicknesses.

The chemical surface maps with average intensities illustrate the coating coverage and homogeneity, where high amounts of free oil result in thick deposits with island-like locations. The thin coatings often result in the formation of striations parallel to the direction of coating, and can mainly be related to local deposits of the organic (imide) coating components. The presentation of free oil in

different amounts depending on the coating type is complementary to the average intensity maps. The chemical surface maps with relative band ratios have been analyzed relative to the organic coating phase and the substrate. The distribution of styrene and imide over the surface agrees fairly well and largely corresponds with the chemical mapping based on average intensities. For coatings with polyunsaturated oils, the surface coverage with high intensities of oil overlaps with the areas of organic coating components, as the oil is largely encapsulated in the organic parts. For coatings with monounsaturated oils, the surface coverage with high intensities of band ratios for oil is complementary to the surface areas with organic coating components, as there is a large amount of free oil as a binder in between the organic coating phase.

In conclusion, the selection of specific oils incorporated in poly(styrene-co-maleic) anhydride allows us to create various distributions of chemical moieties at the surface. Depending on the type of oil chosen, the amount of imide, styrene, and free oil at the coating surface varies. As the resulting hydrophobic properties and barrier performance depend on both the chemical surface composition and topography, the coatings are expected to serve as a candidate for barrier coatings in future packaging applications.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2079-6412/8/5/154/s1, Figur S1: Reference SEM image of uncoated paper and the same magnifications of the coated paper samples; Figures S2–S5: Raman maps (5 × 5 mm2) with single wavenumbers representing different coating and substrate components at the surface for SMI/oil paper coatings, Table S1: Raman shift assignment for SMI/oil coatings on paper.

**Acknowledgments:** The author acknowledges the support of Mr. Peter Mast (Ghent University—Department of Materials Science and Engineering, Belgium) with FEG-SEM. Samples were kindly supplied by Topchim N.V. (Wommelgem, Belgium).

**Conflicts of Interest:** The author declares no conflict of interest.


**Author details**

Priyanka Shanmugavel1 \*, Gowtham Ramasamy2 , Geethalakshmi Vellingiri3 , Rajavel Marimuthu3 and Kalaimagal Thiyagarajan4

1 Department of Plant Breeding and Genetics, Tamil Nadu Agricultural University (TNAU), Coimbatore, India

2 Agro Climate Research Centre, Tamil Nadu Agricultural University, Coimbatore, India

3 Tamil Nadu Agricultural University, Coimbatore, India

4 Department of Millets, Tamil Nadu Agricultural University, Coimbatore, India

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**5. The role of LSCs in tumor proliferation and anti-apoptosis**

Various signaling pathways that stimulate proliferation or inhibit apoptosis are known to aberrant activate LSCs.

#### **5.1 Hedgehog pathway**

The Hh pathway is a highly conserved pathway that regulates the proliferation, migration, and differentiation of cells during development [62, 63]. Three distinct ligands, namely Sonic (Shh), Indian (Ihh), and Desert (Dhh) Hedgehog, exist in humans. Upon ligand binding to the receptor pat (Ptch), inhibition of smoothened (Smo) receptor is relieved. Smo then activates members of the Gli family of zinc-finger transcription factors, translocating them to the nucleus to regulate the transcription of Hh target genes including Gli1, Gli2, and Ptch, and regulators of cell proliferation and survival [64–66].

The Hh pathway promotes cell proliferation mainly by regulating cell cycle. Its regulation mechanism is as follows [67]: (1) Cyclin D1 and cyclin D2 act as downstream target genes for the transcription factor GLI1 and are involved in cell cycle G1 to S phase transformation; (2) PTCH regulates the activity of cyclin B, which is part of the mitosis promoting factor (MPF) compound. MPF is required for cell entry from the G2 phase to the M phase; and (3) SMO proteins block cellular dormancy by modulating P21, a cyclin-dependent inhibitory protein.

The Hh signaling pathway regulates apoptosis mainly through the following mechanisms: (1) Regulate the activity of the BCL-2 family. The BCL-2 family is divided into anti-apoptotic proteins (such as BCL-2, BCL-XL, and MCL-1) and

pro-apoptotic proteins (such as BAX, BAD, and BAK). The ratio between the two types of proteins will directly affect the stability of the mitochondrial membrane and is the most important regulator of the mitochondrial apoptosis pathway. Overexpression of BCL-2, an increase in the ratio of BCL-2 to BAD, leading to defects in mitochondrial apoptosis, is one of the important mechanisms for LSC multidrug resistance and poor prognosis of AML [68]. BCL-2 is the target gene downstream of the Hh pathway, and Hh pathway blockers can induce apoptosis by downregulating BCL-2 [69]. Kobune et al. found that cyclopamine induces apoptosis of drug-resistant CD34+ AML cells by downregulating BCL-2 and makes them sensitive to Ara-C [70]. MCL1 has also emerged as a mechanism of resistance to apoptosis and to BCL-2/BCL-XL inhibitors, and therefore, it is considered as a potential therapeutic target [71]. (2) Regulation of tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) mediated apoptosis. TRAIL-R3 is a blank receptor that lacks a functional death region and is highly expressed in CD34+ CD38<sup>−</sup> LSCs; the downregulation of TRAIL-R3 increases apoptosis [72]. (3) Regulation of FAS protein express in the death receptor pathway. Studies have found that the Hh pathway inhibitor GDC-0449 promotes tumor stem cell apoptosis by upregulating FAS protein [73].


**Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes**

Atsushi Wada *National Defense Academy Japan*

### **1. Introduction**

52 Advanced Holography – Metrology and Imaging

Zhang, T. and Yamaguchi, I. (1998) Three-dimensional microscopy with phase-shifting

Yaroslavskii, L. P. & Merzlyakov, N. S. (1980) *Methods of Digital Holography*, Consultants

Bureau, ISBN<PHONE_NUMBER><PHONE_NUMBER>638, New York

doi:10.1364/OL.23.001221

digital holography, *Optics Letters*, Vol. 23, Issue 15, pp. 1221-1223, ISSN 0146-9592,

Two-wavelength holographic interferometry is an effective technique to generate a contour map of a diffusely reflecting surface (Friesem & Levy (1976); Heflinger & Wuerker (1969); Hildebrand & Haines (1967); Yonemura (1985)). In this technique, two holograms are recorded with two wavelengths. An interference fringe pattern is generated by superposing two object images reconstructed from the holograms.

In digital holography, a hologram is recorded by an image sensor and saved into a computer. An object image can be reconstructed by numerical calculation. Several reconstruction methods were reported. Some of these have adjustablity of position and scale of a reconstruction image (Yu & Kim (2006); Zhang et al. (2004)). An object phase distribution can be obtained by the numerical reconstruction of digital holograms. Therefore, two-wavelength digital holographic interferometry makes it possible to generate a contour map by numerical extraction of a phase difference between two reconstructed images (Gass et al. (2003); Parshall & Kim (2006); Wagner et al. (2000; 1999); Yamaguchi (2001); Yamaguchi et al. (2006)). A phase difference extracted from reconstructed object images is wrapped into a half-open interval (−*π*,*π*]. If a measured object height was large with respect to a synthetic wavelength, 2*π* ambiguities of the phase difference should be eliminated for retrieving the object profile. Common phase unwrapping algorithms (Asundi & Wensen (1998); Servin et al. (1998)) which use phase information of neighbor pixels can be applied when an object structure has no discontinuity. However the algorithms can not work correctly for an object profile having isolated region surrounded by discontinuous step.

An object profile with discontinuous structure can be measured by two-wavelength interferometry with a sufficiently large synthetic wavelength. For example, two-wavelength holographic interferometry with a ruby laser and a synthetic wavelength of ∼ 2 cm was reported (Heflinger & Wuerker (1969); Pedrini et al. (1999)). Nevertheless the measurement error tends to be amplified linearly with an increase in the synthetic wavelength since most of the error sources are the product of the synthetic wavelength and an error of the extracted phase difference (Cheng & Wyant (1984)).

A technique which eliminates 2*π* ambiguities by using a phase difference with a large synthetic wavelength was reported (Cheng & Wyant (1985); de Groot (1991); Wagner et al. (2000)). This technique makes it possible to measure a large step-height with high depth resolution. Wagner *et. al.* reported multiple-wavelength holographic interferometry using a dye laser. They combined three phase differences with synthetic wavelengths of 3.04 mm,

*z*

*xx*� + *f* � *yy*�

)]. (4)

. (5)

*x* = *x*/*λz*

(6)

*x'*

point light source

*h* θ

Fig. 1. Geometry of holographic interferometry.

Equation (2) can be written in terms of *U*�

*y*) = F(*u*�

*u*o(*x*, *y*, *z*) = 1

*I*(*x*, *y*, *z*) = |*u*r(*x*, *y*, *z*) + *u*o(*x*, *y*, *z*)|

2 + |*u*o|

= |*u*r|

= 1 *z*2 1 + *U*� o *x λz* , *y λz* 2 + *U*� o *x λz* , *y λz* + *U*�∗ o *x λz* , *y λz*

= 1 *z*2 1 + *U*� o(*f* � *x*, *f* � *y*) 2 + *U*� o(*f* � *x*, *f* �

o) = *dx*�

2 + *u*∗

*z* exp

*dy*� *u*� o(*x*� , *y*�

*iπ*

An intensity distribution *I*(*x*, *y*, *z*) of an interference fringe pattern formed by the object wave

2

In Eq. (6), the third term includes the Fourier spectrum of the object light. The object image can be obtained by applying an inverse Fourier transform to *I*(*x*, *y*, *z*). The reconstructed wave

o

r *u*o + *u*r*u*∗

*x*2 + *y*2 *λz*

 *U*� o *x λz* , *y λz*

*y*) + *U*�∗ o (*f* � *x*, *f* � *y*) .

*U*� o(*f* � *x*, *f* �

and the reference wave is given by

*y* = *y*/*λz*:

and

and *f* �

reference wave

3D object image sensor

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 55

object wave

illumination light

) exp[−*i*2*π*(*f* �

o by substituting the spatial frequencies *f* �

1.53 mm and 0.76 mm, and realized the measurement with a measurable step-height of 0.6 mm and an error of 36 *μ*m (Wagner et al. (2000)).

In this paper, multiple-wavelength digital holographic interferometry using tunablity of laser diodes for measuremnt of a large step-height with high accuracy is presented. Using the high-resolution wavelength tunability of laser diodes, a pair of holograms with a small wavelength difference less than 0.01 nm can be recorded, and used to extract a phase difference with a large synthetic wavelength of more than 120 mm. Several holograms are recorded through the change in the wavelength of a laser diode. Phase differences with synthetic wavelengths from 0.4637 mm to 129.1 mm are extracted from the holograms. The synthetic wavelength of 129.1 mm allows us to measure a step-height of 32 mm. The synthetic wavelength of 0.463 mm presents us with a measurement with an rms error of 0.01 mm. 2*π* ambiguities of the phase difference with a small synthetic wavelength are eliminated by a recursive calculations using the phase difference with a larger synthetic wavelength. The elimination of 2*π* ambiguities realizes the measurement with the measurable step-height of 32 mm and the rms error of 0.01. The requirements for performing the phase unwrapping are discussed. We found that precise knowledge of the recording wavelengths is required for correctly performing the phase unwrapping. The required precision of the knowledge is derived.

The Fresnel-transform based hologram reconstruction is fast and commonly used. However, the pixel size of reconstructed image increases with the recording wavelength. If the pixel size variation becomes large, the extraction of a phase difference cannot be performed correctly. In order to obtain a correct phase difference, the pixel size should be adjusted. A simple and fast algorithm for pixel size adjustment is described.

### **2. Hologram recording and reconstruction**

Figure 1 shows a geometry of holographic interferometry. A setup for off-axis lensless Fourier transform holography is used for hologram recording. A spherical wave from a point light source is used as a reference wave. An object is located in the same plane as the point light source. A distance between the object and a hologram satisfies the Fresnel approximation (Goodman (1968)). Let the point light source be at *x* = 0, *y* = 0, *z* = 0 and the reference wave *u*r(*x*, *y*, *z*) is given by

$$u\_{\mathbf{r}}(\mathbf{x}, y, z) = \frac{1}{z} \exp\left(i\pi \frac{\mathbf{x}^2 + y^2}{\lambda z}\right),\tag{1}$$

where *λ* is wavelength.

Let *u*o(*x*, *y*, *z*) is a complex amplitude of the object light at the hologram. By using a complex amplitude distribution *u*o(*x*� , *y*� , 0) of the object light in the plane of *z* = 0, *u*o(*x*, *y*, *z*) is given by

$$u\_0(\mathbf{x}, y, z) = \frac{1}{z} \exp\left(i\pi \frac{\mathbf{x}^2 + y^2}{\lambda z}\right) \iint d\mathbf{x}' dy' u\_0(\mathbf{x}', y', 0) \exp\left\{i2\pi \left(\frac{\mathbf{x}'^2 + y'^2}{2\lambda z} - \frac{\mathbf{x}\mathbf{x}' + y\mathbf{y}'}{\lambda z}\right)\right\}.\tag{2}$$

We define *u*� o as the product of *u*o(*x*� , *y*� , 0) and a spherical wave phase factor, and *U*� o as the Fourier spectrum of *u*� o:

$$u\_o'(x',y') = u\_o(x',y',0) \exp\left(i\pi \frac{x'^2 + y'^2}{\lambda z}\right),\tag{3}$$

Fig. 1. Geometry of holographic interferometry.

and

2 Hologram / Book 2

1.53 mm and 0.76 mm, and realized the measurement with a measurable step-height of 0.6

In this paper, multiple-wavelength digital holographic interferometry using tunablity of laser diodes for measuremnt of a large step-height with high accuracy is presented. Using the high-resolution wavelength tunability of laser diodes, a pair of holograms with a small wavelength difference less than 0.01 nm can be recorded, and used to extract a phase difference with a large synthetic wavelength of more than 120 mm. Several holograms are recorded through the change in the wavelength of a laser diode. Phase differences with synthetic wavelengths from 0.4637 mm to 129.1 mm are extracted from the holograms. The synthetic wavelength of 129.1 mm allows us to measure a step-height of 32 mm. The synthetic wavelength of 0.463 mm presents us with a measurement with an rms error of 0.01 mm. 2*π* ambiguities of the phase difference with a small synthetic wavelength are eliminated by a recursive calculations using the phase difference with a larger synthetic wavelength. The elimination of 2*π* ambiguities realizes the measurement with the measurable step-height of 32 mm and the rms error of 0.01. The requirements for performing the phase unwrapping are discussed. We found that precise knowledge of the recording wavelengths is required for correctly performing the phase unwrapping. The required precision of the knowledge is

The Fresnel-transform based hologram reconstruction is fast and commonly used. However, the pixel size of reconstructed image increases with the recording wavelength. If the pixel size variation becomes large, the extraction of a phase difference cannot be performed correctly. In order to obtain a correct phase difference, the pixel size should be adjusted. A simple and fast

Figure 1 shows a geometry of holographic interferometry. A setup for off-axis lensless Fourier transform holography is used for hologram recording. A spherical wave from a point light source is used as a reference wave. An object is located in the same plane as the point light source. A distance between the object and a hologram satisfies the Fresnel approximation (Goodman (1968)). Let the point light source be at *x* = 0, *y* = 0, *z* = 0 and the reference wave

*z* exp

Let *u*o(*x*, *y*, *z*) is a complex amplitude of the object light at the hologram. By using a complex

 *iπ* *x*2 + *y*2 *λz*

, 0) exp

, 0) of the object light in the plane of *z* = 0, *u*o(*x*, *y*, *z*) is given

, 0) and a spherical wave phase factor, and *U*�

*x*�2 + *y*�2 *λz*

*x*�2 + *y*�2

 *i*2*π*

, (1)

2*λz* − *xx*� + *yy*�

, (3)

*λz*

 . (2)

o as the

*u*r(*x*, *y*, *z*) = 1

, *y*�

*dx*� *dy*� *u*o(*x*� , *y*�

, *y*�

) = *u*o(*x*�

, *y*�

, 0) exp

 *iπ*

*x*2 + *y*2 *λz*

o as the product of *u*o(*x*�

*u*� o(*x*� , *y*�

o:

mm and an error of 36 *μ*m (Wagner et al. (2000)).

algorithm for pixel size adjustment is described.

**2. Hologram recording and reconstruction**

derived.

*u*r(*x*, *y*, *z*) is given by

where *λ* is wavelength.

*u*o(*x*, *y*, *z*) = 1

We define *u*�

Fourier spectrum of *u*�

by

amplitude distribution *u*o(*x*�

*z* exp

 *iπ*

$$\mathcal{U}\_{\mathbf{o}}'(f\_{\mathbf{x'}}'f\_{\mathbf{y}}') = \mathcal{F}(\mathbf{u}\_{\mathbf{o}}') = \iint d\mathbf{x'}d\mathbf{y'} \mathbf{u}\_{\mathbf{o}}'(\mathbf{x'},\mathbf{y'}) \exp[-i2\pi(f\_{\mathbf{x}}'\mathbf{x'} + f\_{\mathbf{y}}'\mathbf{y'})].\tag{4}$$

Equation (2) can be written in terms of *U*� o by substituting the spatial frequencies *f* � *x* = *x*/*λz* and *f* � *y* = *y*/*λz*:

$$u\_0(x,y,z) = \frac{1}{z} \exp\left(i\pi \frac{x^2 + y^2}{\lambda z}\right) \mathcal{U}\_\mathbf{o}'\left(\frac{x}{\lambda z}, \frac{y}{\lambda z}\right). \tag{5}$$

An intensity distribution *I*(*x*, *y*, *z*) of an interference fringe pattern formed by the object wave and the reference wave is given by

$$\begin{split} I(\mathbf{x}, y, z) &= |u\_{\mathbf{r}}(\mathbf{x}, y, z) + u\_{\mathbf{o}}(\mathbf{x}, y, z)|^2 \\ &= |u\_{\mathbf{r}}|^2 + |u\_{\mathbf{o}}|^2 + u\_{\mathbf{r}}^\* u\_{\mathbf{o}} + u\_{\mathbf{r}} u\_{\mathbf{o}}^\* \\ &= \frac{1}{z^2} \left\{ 1 + \left| \mathcal{U}\_{\mathbf{o}}' \left( \frac{\mathbf{x}}{\lambda z'}, \frac{y}{\lambda z} \right) \right|^2 + \mathcal{U}\_{\mathbf{o}}' \left( \frac{\mathbf{x}}{\lambda z'}, \frac{y}{\lambda z} \right) + \mathcal{U}\_{\mathbf{o}}'^\* \left( \frac{\mathbf{x}}{\lambda z}, \frac{y}{\lambda z} \right) \right\} \\ &= \frac{1}{z^2} \left\{ 1 + \left| \mathcal{U}\_{\mathbf{o}}' (f\_{\mathbf{x}'}' f\_{\mathbf{y}}') \right|^2 + \mathcal{U}\_{\mathbf{o}}' (f\_{\mathbf{x}'}' f\_{\mathbf{y}}') + \mathcal{U}\_{\mathbf{o}}'^\* (f\_{\mathbf{x}'}' f\_{\mathbf{y}}') \right\}. \end{split} \tag{6}$$

In Eq. (6), the third term includes the Fourier spectrum of the object light. The object image can be obtained by applying an inverse Fourier transform to *I*(*x*, *y*, *z*). The reconstructed wave

where *L* is an optical path difference caused by the object surface structure, giving

, *y*�

Holograms are recorded with wavelengths of *λn* satisfying *λn* < *λn*+1 and the phase distribution of the object wave at the plane of *z* = 0 is *φn*. The phase difference Δ*φn* between

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 57

Λ*n*

In Eq. (8), the condition *z* � (*x*�2 + *y*�2)/Λ*n* is assumed and a quadratic phase term is

Let Ψ*n* be the phase difference extracted from the image reconstructed from the hologram

Re(*u*� o*nu*� o0 ∗ )

where *mn* is an integer. If the conditions of Δ*φ*1 = Ψ1 and Λ*n* < Λ*n*−1 are satisfied, Δ*φn* can be retrieved (Nadeborn et al. (1996); Paulsson et al. (2000); Wagner et al. (2000)) through

Δ*φn* = Ψ*n* + 2*π*NINT *αn*Δ*φn*−1 − Ψ*n*

where NINT(*a*) denotes the function returning the nearest neighbor integer of argument *a*, and *αn* is the sensitivity ratio between each phase difference Δ*φn* and Δ*φn*−1 and given by

> *αn* = Λ*n*−1 Λ*n*

Ψ*n* = tan−1 Im(*u*�

Ψ*n* is wrapped into (−*π*,*π*] and the relation between Ψ*n* and Δ*φn* is given by

 *L* +

> � *λ*¯ *n* 2 Δ*λn*

> > o*nu*� o0 ∗ )

> > > 2*π*

*x*�2 + *y*�2 2*z*

 � 2*π* Λ*n*

2 , Δ*λn* = *λn* − *λ*0. (16)

Δ*φn* = Ψ*n* + 2*πmn*, (18)

) − *x*� sin *θ*. (13)

*L*, (14)

, (15)

. (17)

, (19)

. (20)

*L* = (1 + cos *θ*)*h*(*x*�

The second term of Eq. (12) is the spherical wavefront introduced in Eq. (3).

) = *φn* − *φ*0 = 2*π*

Λ*n* = *λ*0*λn λn* − *λ*0

where *λ*¯ *n* and Δ*λn* are the average and difference of wavelengths given by

*λ*¯ *n* = *λ*0 + *λn*

with wavelength of *λn*. The phase difference Ψ*n* are given by

*φn* and *φ*0 is given by

neglected.

recursive calculation of

Δ*φn*(*x*�

where Λ*n* are synthetic wavelengths and

, *y*�

field *u*rec obtained by the transformation is given by

$$\begin{split} u\_{\text{rec}}(\mathbf{x}',y') &= \mathcal{F}^{-1}\{I(\mathbf{x},y,z)\} \\ &= \mathcal{F}^{-1}\{I(\lambda z f\_{\mathbf{x}'}^{\prime}\lambda z f\_{y^{\prime}}^{\prime}z)\} \\ &= \iint df\_{\mathbf{x}}^{\prime} df\_{\mathbf{y}}^{\prime} \, I(\lambda z f\_{\mathbf{x}'}^{\prime}\lambda z f\_{y^{\prime}}^{\prime}z) \exp[i2\pi(f\_{\mathbf{x}}^{\prime}\mathbf{x}' + f\_{\mathbf{y}}^{\prime}y')] \\ &= \lambda^{2} \left\{\frac{\delta(\mathbf{x}',y^{\prime})}{\lambda^{2}z^{2}} + u\_{\mathbf{o}}^{\prime}(\mathbf{x}',y^{\prime}) \otimes u\_{\mathbf{o}}^{\prime \*}(-\mathbf{x}',-y^{\prime}) + u\_{\mathbf{o}}^{\prime}(\mathbf{x}',y^{\prime}) + u\_{\mathbf{o}}^{\prime \*}(-\mathbf{x}',-y^{\prime})\right\}, \end{split} \tag{7}$$

where the symbol ⊗ stands for the convolution operator. In Eq. (7), the first term is the reference point light source. The second term is an autocorrelation of the object light. The third term expresses the object light produced by a spherical wave phase factor. The fourth term is the complex conjugate of the third term. Since a phase factor does not affect intensity, the intensity distribution of the object light can be reconstructed by the third term of Eq. (7). Let consider that *I*(*x*, *y*, *z*) is recorded by an image sensor having *Nx* × *Ny* pixels with pixel size of Δ*x* × Δ*y*. Numerical reconstruction of the digital hologram is realized through the inverse discrete Fourier transform. Let the pixel size of the numerically reconstructed object image be Δ*x* � × Δ*y* � . The discrete formulation of Eq. (7) is then

$$u\_{\rm PC}(s\Delta\_{\rm x}^{\prime}, t\Delta\_{\rm y}^{\prime}) = \sum\_{p=-N\_{\rm x}/2}^{N\_{\rm x}/2-1} \sum\_{q=-N\_{\rm y}/2}^{N\_{\rm y}/2-1} I(p\Delta\_{\rm x}, q\Delta\_{\rm y}, z) \exp\left\{i2\pi \left(\frac{p\Delta\_{\rm x}}{\lambda z}s\Delta\_{\rm x}^{\prime} + \frac{q\Delta\_{\rm y}}{\lambda z}t\Delta\_{\rm y}^{\prime}\right)\right\},\tag{8}$$

where *p*, *q* are integers, and *s*, *t* are integers and fulfill *x*� = *s*Δ*x* � , *y*� = *t*Δ*y* � . The right side of Eq. (8) is arranged in the form, in which fast Fourier transform can be applied:

$$\mu\_{\text{rec}}(\text{s}\Delta\_{\text{x}}^{\prime}, t\Delta\_{\text{y}}^{\prime}) = \sum\_{p=-N\_{\text{x}}/2}^{N\_{\text{x}}/2-1} \sum\_{q=-N\_{\text{y}}/2}^{N\_{\text{y}}/2-1} I(p\Delta\_{\text{x}}, q\Delta\_{\text{y}}, z) \exp\left\{i2\pi \left(\frac{p\text{s}}{N\_{\text{x}}} + \frac{qt}{N\_{\text{y}}}\right)\right\},\tag{9}$$

where *s*, *t*, *p* and *q* are integers, and Δ� *x*, Δ� *y* satisfy the following condition

$$
\Delta'\_{\mathbf{x}} = \frac{\lambda z}{\Delta\_{\mathbf{x}} \mathbf{N}\_{\mathbf{x}}}, \ \Delta'\_{\mathbf{y}} = \frac{\lambda z}{\Delta\_{\mathbf{y}} \mathbf{N}\_{\mathbf{y}}}, \tag{10}
$$

and

$$-N\_{\rm x}/2 \le s < N\_{\rm x}/2, \ -N\_{\rm y}/2 \le t < N\_{\rm y}/2. \tag{11}$$

The field of view of the reconstructed image is *λz*/Δ*x* × *λz*/Δ*y*.

#### **3. Multiple-wavelength interferometry**

The reference plane is assumed to be placed at *z* = 0. Let the illumination light irradiate to the object at an angle *θ* to the reference plane, and the height of the object surface respect to the reference plane be *h*(*x*� , *y*� ). The phase distribution *φ*(*x*� , *y*� ) of the reconstructed image is given by

$$\phi(\mathbf{x'}, \mathbf{y'}) = \frac{2\pi}{\lambda} \left\{-L + \frac{\mathbf{x'^2} + \mathbf{y'^2}}{2z}\right\},\tag{12}$$

where *L* is an optical path difference caused by the object surface structure, giving

$$L = (1 + \cos \theta) h(x', y') - x' \sin \theta. \tag{13}$$

The second term of Eq. (12) is the spherical wavefront introduced in Eq. (3). Holograms are recorded with wavelengths of *λn* satisfying *λn* < *λn*+1 and the phase distribution of the object wave at the plane of *z* = 0 is *φn*. The phase difference Δ*φn* between *φn* and *φ*0 is given by

$$
\Delta\phi\_{\rm n}(\mathbf{x}', \mathbf{y}') = \phi\_{\rm n} - \phi\_0 = \frac{2\pi}{\Lambda\_{\rm n}} \left( L + \frac{\mathbf{x}'^2 + \mathbf{y}'^2}{2z} \right) \simeq \frac{2\pi}{\Lambda\_{\rm n}} L,\tag{14}
$$

where Λ*n* are synthetic wavelengths and

4 Hologram / Book 2

*y*, *z*) exp[*i*2*π*(*f* �

o (−*x*�

where the symbol ⊗ stands for the convolution operator. In Eq. (7), the first term is the reference point light source. The second term is an autocorrelation of the object light. The third term expresses the object light produced by a spherical wave phase factor. The fourth term is the complex conjugate of the third term. Since a phase factor does not affect intensity, the intensity distribution of the object light can be reconstructed by the third term of Eq. (7). Let consider that *I*(*x*, *y*, *z*) is recorded by an image sensor having *Nx* × *Ny* pixels with pixel size of Δ*x* × Δ*y*. Numerical reconstruction of the digital hologram is realized through the inverse discrete Fourier transform. Let the pixel size of the numerically reconstructed object

*I*(*p*Δ*x*, *q*Δ*y*, *z*) exp

, Δ�

The reference plane is assumed to be placed at *z* = 0. Let the illumination light irradiate to the object at an angle *θ* to the reference plane, and the height of the object surface respect to

). The phase distribution *φ*(*x*�

) = 2*π λ* −*L* +

) ⊗ *u*�∗

. The discrete formulation of Eq. (7) is then

Eq. (8) is arranged in the form, in which fast Fourier transform can be applied:

*Ny*/2−1 ∑ *q*=−*Ny*/2

*x*, Δ�

*x* = *λz* Δ*xNx*

*Ny*/2−1 ∑ *q*=−*Ny*/2

where *p*, *q* are integers, and *s*, *t* are integers and fulfill *x*� = *s*Δ*x*

*Nx*/2−1 ∑ *p*=−*Nx*/2

Δ�

The field of view of the reconstructed image is *λz*/Δ*x* × *λz*/Δ*y*.

, *y*�

*φ*(*x*� , *y*� *xx*� + *f* � *yy*� )]

> ) + *u*� o(*x*� , *y*�

 *i*2*π*

*I*(*p*Δ*x*, *q*Δ*y*, *z*) exp

*y* = *λz* Δ*yNy*

*y* satisfy the following condition

−*Nx*/2 ≤ *s* < *Nx*/2, −*Ny*/2 ≤ *t* < *Ny*/2. (11)

, *y*�

*x*�2 + *y*�2 2*z*

 *p*Δ*x λz s*Δ�

�

 *i*2*π*

, *y*� = *t*Δ*y*

 *ps Nx*

) + *u*�∗

o (−*x*�

*x* + *q*Δ*y λz t*Δ� *y*

�

+ *qt Ny*

, (10)

) of the reconstructed image is

, (12)

, −*y*� ) , (7)

, (8)

, (9)

. The right side of

, −*y*�

field *u*rec obtained by the transformation is given by

*Nx*/2−1 ∑ *p*=−*Nx*/2

*x*, *λz f* �

*y I*(*λz f* �

*y*, *z*)}

*x*, *λz f* �

o(*x*� , *y*�

) =F −1{*I*(*x*, *y*, *z*)} =F −1{*I*(*λz f* �

> *δ*(*x*� , *y*� ) *λ*2*z*2 + *u*�

= *d f* � *xd f* �

=*λ*2

� × Δ*y* �

*u*rec(*s*Δ*x* � , *t*Δ*y* � ) =

where *s*, *t*, *p* and *q* are integers, and Δ�

**3. Multiple-wavelength interferometry**

the reference plane be *h*(*x*�

*u*rec(*x*� , *y*�

image be Δ*x*

*u*rec(*s*Δ*x* � , *t*Δ*y* � ) =

and

given by

$$
\Lambda\_{\rm ll} = \frac{\lambda\_0 \lambda\_{\rm ll}}{\lambda\_{\rm ll} - \lambda\_0} \simeq \frac{\bar{\lambda\_{\rm ll}}^2}{\Delta \lambda\_{\rm ll}},
\tag{15}
$$

where *λ*¯ *n* and Δ*λn* are the average and difference of wavelengths given by

$$
\tilde{\lambda}\_{\mathfrak{n}} = \frac{\lambda\_0 + \lambda\_{\mathfrak{n}}}{2}, \,\Delta\lambda\_{\mathfrak{n}} = \lambda\_{\mathfrak{n}} - \lambda\_{\mathfrak{d}}.\tag{16}
$$

In Eq. (8), the condition *z* � (*x*�2 + *y*�2)/Λ*n* is assumed and a quadratic phase term is neglected.

Let Ψ*n* be the phase difference extracted from the image reconstructed from the hologram with wavelength of *λn*. The phase difference Ψ*n* are given by

$$\Psi\_{\rm II} = \tan^{-1} \frac{\mathrm{Im}(\mu\_{\rm on}' \mu\_{\rm o0}'^\*)}{\mathrm{Re}(\mu\_{\rm on}' \mu\_{\rm o0}'^\*)}.\tag{17}$$

Ψ*n* is wrapped into (−*π*,*π*] and the relation between Ψ*n* and Δ*φn* is given by

$$
\Delta \phi\_{\rm ll} = \Psi\_{\rm ll} + 2\pi m\_{\rm ll} \tag{18}
$$

where *mn* is an integer. If the conditions of Δ*φ*1 = Ψ1 and Λ*n* < Λ*n*−1 are satisfied, Δ*φn* can be retrieved (Nadeborn et al. (1996); Paulsson et al. (2000); Wagner et al. (2000)) through recursive calculation of

$$
\Delta\phi\_{\rm nl} = \Psi\_{\rm nl} + 2\pi \text{NINT}\left(\frac{\alpha\_{\rm nl}\Delta\phi\_{\rm nl-1} - \Psi\_{\rm nl}}{2\pi}\right),
\tag{19}
$$

where NINT(*a*) denotes the function returning the nearest neighbor integer of argument *a*, and *αn* is the sensitivity ratio between each phase difference Δ*φn* and Δ*φn*−1 and given by

$$
\alpha\_{\rm ll} = \frac{\Lambda\_{\rm n-1}}{\Lambda\_{\rm n}}.\tag{20}
$$

and

Λ�

The requirements of *�* and *α*�

*n* = 2.5 mm and *L* = 25 mm.

wavelength. For example, if *α*�

propagation of Δ*λn* in Eq. (15), we have

Let the maximum of |Δ*λ�*,*n*| for all *n* be Δ*λ�* :

*n* = 2, *λ*¯

**5. Scale adjustment for phase calculation**

should be less than 1/20.

For example, *α*�

nm.

*α*� *n* <

in previous works (Nadeborn et al. (1996); Wagner et al. (2000)). For example, if *α*�

the upper limit of *�* is 1/6. In other words, if *�* = 1/12 then the upper limit of *α*�

for the phase unwrapping. By solving Eq. (28) for |*α�*,*n*| , we obtain the condition

*n* Λ� *n L*

performing the phase unwrapping become small in proportion to *α*�

*n* = 2, Λ�

phase unwrapping. Let the true value of wavelength be *λ*�

*α�*,*n α*� *n*

*L* Λ� *n* =

Δ*λ�* <

*λ*¯ *n* 2 *L*

error of the wavelength difference are defined as Δ*λ*�

*n* = 1/40, the upper limit of *�* and *α*�

1 − 2*�*

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 59

be noted that the requirements denoted in Eqs. (29) and (30) are based on the assumption of

Let us consider how accurate an estimation of the phase difference sensitivity ratio is required

1

This shows that the upper limit of the measurement error of the sensitivity ratio for correctly

*αn* should be precisely estimated for measuring a large object height by using a small synthetic

Let us consider how precise the measurement of the wavelengths is required to be for the

Δ*λ�*,*n*−1 Δ*λ*� *n*−1

By substituting Eqs. (32) and (33) into Eq. (28) and solving for Δ*λ�*, we obtain the condition

 1 2(*α*�

In the Fresnel-transform based method (FTM), the pixel size of the reconstructed image increases with the reconstruction distance and the recording wavenumber. The variation of the pixel size poses problems in holographic interferometry (Yamaguchi et al. (2006)). In contrast, the convolution method (CM) (Yamaguchi et al. (2002)) keeps the pixel size of the reconstructed image the same as the pixel size of an image sensor used for hologram recording. However, the CM requires huge computational time with respect to the FTM. A

− Δ*λ�*,*n* Δ*λ*� *n*

*n* + 1) − *�*

*n* = 800 nm, *L* =25 mm, and *�* = 1/12, Δ*λ�* should be less than 0.002

 *L*Δ*λ*� *n λ*¯ *n*

*n*. If there has been a small error of the phase difference sensitivity ratio such as

2 − *�*(*α*�

*n* + 1)

2*�* . (30)

*n* become small to 1/12 and 2 respectively with

*n*Λ�

*n* = 2.5 mm, *L* = 25 mm and *�* = 1/12 , then |*α�*,*n*|

. (31)

*n*, and the true and the measurement

2 . (32)

. (34)

*n* and Δ*λ�*,*n*. By calculating the error

*n*/*L*. This means that

*n* = 2 then

*n* is 5. It should

*n* seem to be relaxed as compared with the requirements derived

#### **4. Requirements for the phase unwrapping**

In this section, we discuss the requirements for correctly performing the phase unwrapping. First, the requirements of the measurement error and the sensitivity ratio between the phase differences are discussed based on the assumption that the error of the sensitivity ratio is negligible small. Next, it is pointed out that the sensitivity ratio should be estimated for the phase unwrapping. Then, the required accuracy of the sensitivity ratio is derived. Lastly, it is explained how precise measurement of the wavelengths is required for high accuracy estimation of the sensitivity ratio.

Let the true phase differences and the sensitivity ratio are defined as Δ*φ*� *n*, Ψ� *n* and *α*� *n*, and the measurement errors of the phase differences and the sensitivity ratio are defined as Ψ*�*,*n* and *α�*,*n*. The relations between these parameters and Ψ, Δ*φ* and *α* are given by

$$
\Delta \mathbf{V}\_{\rm n} = \mathbf{V}\_{\rm n}^{\prime} + \mathbf{V}\_{\varepsilon, \rm n} \cdot \Delta \boldsymbol{\phi}\_{\rm n}^{\prime} = \mathbf{V}\_{\rm n}^{\prime} + 2\pi m\_{\rm n} = \Delta \boldsymbol{\phi}\_{\rm n} - \mathbf{V}\_{\varepsilon, \rm n} \cdot \mathbf{a}\_{\rm n} = \mathbf{a}\_{\rm n}^{\prime} + \mathbf{a}\_{\varepsilon, \rm n}. \tag{21}
$$

By substituting Eqs. (21) into Eq. (19), we get

$$\Delta\phi\_{\rm n} = \mathbf{Y}\_{\rm n}^{\prime} + \mathbf{Y}\_{\varepsilon,\rm n} + 2\pi \text{NINT}\left(\frac{(\mathbf{a}\_{\rm n}^{\prime} + \mathbf{a}\_{\varepsilon,\rm n})(\Delta\phi\_{\rm n-1}^{\prime} + \mathbf{Y}\_{\varepsilon,\rm n-1}) - \mathbf{Y}\_{\rm n}^{\prime} - \mathbf{Y}\_{\varepsilon,\rm n}}{2\pi}\right). \tag{22}$$

For the true phase difference and sensitivity ratio, the condition

$$\frac{\alpha\_n' \Delta \phi\_{n-1}' - \Psi\_n'}{2\pi} = \frac{\Delta \phi\_n' - \Psi\_n'}{2\pi} = m\_{n\nu} \tag{23}$$

is satisfied. Substituting Eqs. (23) into Eq. (22) gives

$$
\Delta\phi\_{\rm nl} = \Delta\phi\_{\rm nl}^{\prime} + \Psi\_{\varepsilon,\rm ll} + 2\pi\text{NINT}\left(\frac{a\_{\varepsilon,\rm ll}(\Delta\phi\_{\rm n-1}^{\prime} + \Psi\_{\varepsilon,\rm n-1}) + a\_{\rm ll}\Psi\_{\varepsilon,\rm n-1} - \Psi\_{\varepsilon,\rm n}}{2\pi}\right). \tag{24}
$$

In order to correctly perform the phase unwrapping, the third term in the right side of Eq. (24) should be zero, and the condition

$$\left|\frac{a\_{\varepsilon,n}\Delta\phi\_{n-1}^{\prime} + a\_n\Psi\_{\varepsilon,n-1} - \Psi\_{\varepsilon,n}}{2\pi}\right| < \frac{1}{2},\tag{25}$$

should be satisfied. In Eq. (25), the condition Ψ*�*,*n*−1 � Δ*φ*� *n*−1 is assumed. Let the maximum of |Ψ*�*,*n*| for all *n* be 2*π�*:

$$|\Psi\_{\epsilon,\mathfrak{u}}| \le 2\pi\epsilon.\tag{26}$$

Let the true synthetic wavelength be defined as Λ� *n*. Substituting Δ*φ*� *n*−1, *α*� *n* and Λ� *n* into Eq. (8) gives

$$
\Delta \phi\_{n-1}' = \frac{2\pi L}{\alpha\_n' \Lambda\_n'}.\tag{27}
$$

Substitution of Eqs. (26) and (27) into Eq. (25) gives

$$
\epsilon < \frac{1}{a\_n' + 1} \left( \frac{1}{2} - \frac{|\alpha\_{\varepsilon, n}|}{a\_n'} \frac{L}{\Lambda\_n'} \right). \tag{28}
$$

If the assumption |*α�*,*n*| � *α*� *n* is introduced, we have the requirements

$$
\epsilon < \frac{1}{2(\alpha\_n' + 1)},\tag{29}
$$

and

6 Hologram / Book 2

In this section, we discuss the requirements for correctly performing the phase unwrapping. First, the requirements of the measurement error and the sensitivity ratio between the phase differences are discussed based on the assumption that the error of the sensitivity ratio is negligible small. Next, it is pointed out that the sensitivity ratio should be estimated for the phase unwrapping. Then, the required accuracy of the sensitivity ratio is derived. Lastly, it is explained how precise measurement of the wavelengths is required for high accuracy

measurement errors of the phase differences and the sensitivity ratio are defined as Ψ*�*,*n* and

*n* + *α�*,*n*)(Δ*φ*�

*n* + 2*πmn* = Δ*φn* − Ψ*�*,*n*, *αn* = *α*�

*n* − Ψ� *n*

*n*−1 + Ψ*�*,*n*−1) − Ψ�

*n*−1 + Ψ*�*,*n*−1) + *αn*Ψ*�*,*n*−1 − Ψ*�*,*n* 2*π*

 < 1 2

*n*. Substituting Δ*φ*�

*L* Λ� *n*

2*π*

*n*, Ψ�

*n* and *α*�

*n* − Ψ*�*,*n*

2*π* = *mn*, (23)

*n* + *α�*,*n*. (21)

, (25)

*n* and Λ�

*n*−1 is assumed. Let the maximum

*n*−1, *α*�

. (27)

, (29)

. (28)

*n*, and the

. (22)

. (24)

*n* into Eq.

Let the true phase differences and the sensitivity ratio are defined as Δ*φ*�

*α�*,*n*. The relations between these parameters and Ψ, Δ*φ* and *α* are given by

*n* = Ψ�

(*α*�

*n*−1 − Ψ� *n* 2*π* = Δ*φ*�

Δ*φ*�

1 *α*� *n* + 1

*�* <

*α�*,*n*(Δ*φ*�

In order to correctly perform the phase unwrapping, the third term in the right side of Eq. (24)

*n*−1 + *αn*Ψ*�*,*n*−1 − Ψ*�*,*n* 2*π*

> *n*−1 = 2*πL α*� *n*Λ� *n*

> > 2 − |*α�*,*n*| *α*� *n*

*n* is introduced, we have the requirements

1 2(*α*� *n* + 1)

1

**4. Requirements for the phase unwrapping**

estimation of the sensitivity ratio.

Ψ*n* = Ψ�

Δ*φn* = Ψ�

Δ*φn* = Δ*φ*�

of |Ψ*�*,*n*| for all *n* be 2*π�*:

If the assumption |*α�*,*n*| � *α*�

(8) gives

should be zero, and the condition

*n* + Ψ*�*,*n*, Δ*φ*�

*n* + Ψ*�*,*n* + 2*π*NINT

is satisfied. Substituting Eqs. (23) into Eq. (22) gives

For the true phase difference and sensitivity ratio, the condition

*α*� *n*Δ*φ*�

*n* + Ψ*�*,*n* + 2*π*NINT

should be satisfied. In Eq. (25), the condition Ψ*�*,*n*−1 � Δ*φ*�

Let the true synthetic wavelength be defined as Λ�

Substitution of Eqs. (26) and (27) into Eq. (25) gives

*α�*,*n*Δ*φ*�

*�* <

By substituting Eqs. (21) into Eq. (19), we get

$$
\alpha'\_n < \frac{1 - 2\varepsilon}{2\varepsilon}.\tag{30}
$$

The requirements of *�* and *α*� *n* seem to be relaxed as compared with the requirements derived in previous works (Nadeborn et al. (1996); Wagner et al. (2000)). For example, if *α*� *n* = 2 then the upper limit of *�* is 1/6. In other words, if *�* = 1/12 then the upper limit of *α*� *n* is 5. It should be noted that the requirements denoted in Eqs. (29) and (30) are based on the assumption of |*α�*,*n*| � *α*� *n*. If there has been a small error of the phase difference sensitivity ratio such as |*α�*,*n*|/*α*� *n* = 1/40, the upper limit of *�* and *α*� *n* become small to 1/12 and 2 respectively with Λ� *n* = 2.5 mm and *L* = 25 mm.

Let us consider how accurate an estimation of the phase difference sensitivity ratio is required for the phase unwrapping. By solving Eq. (28) for |*α�*,*n*| , we obtain the condition

$$|a\_{\varepsilon,n}| < \alpha\_n' \frac{\Lambda\_n'}{L} \left(\frac{1}{2} - \epsilon(\alpha\_n' + 1)\right). \tag{31}$$

This shows that the upper limit of the measurement error of the sensitivity ratio for correctly performing the phase unwrapping become small in proportion to *α*� *n*Λ� *n*/*L*. This means that *αn* should be precisely estimated for measuring a large object height by using a small synthetic wavelength. For example, if *α*� *n* = 2, Λ� *n* = 2.5 mm, *L* = 25 mm and *�* = 1/12 , then |*α�*,*n*| should be less than 1/20.

Let us consider how precise the measurement of the wavelengths is required to be for the phase unwrapping. Let the true value of wavelength be *λ*� *n*, and the true and the measurement error of the wavelength difference are defined as Δ*λ*� *n* and Δ*λ�*,*n*. By calculating the error propagation of Δ*λn* in Eq. (15), we have

$$\frac{\alpha\_{\varepsilon,\boldsymbol{n}}}{\alpha\_{\boldsymbol{n}}^{\prime}} \frac{L}{\Lambda\_{\boldsymbol{n}}^{\prime}} = \left(\frac{\Delta\lambda\_{\varepsilon,\boldsymbol{n}-1}}{\Delta\lambda\_{\boldsymbol{n}-1}^{\prime}} - \frac{\Delta\lambda\_{\varepsilon,\boldsymbol{n}}}{\Delta\lambda\_{\boldsymbol{n}}^{\prime}}\right) \frac{L\Delta\lambda\_{\boldsymbol{n}}^{\prime}}{\bar{\lambda}\_{\boldsymbol{n}}^{\prime}}.\tag{32}$$

Let the maximum of |Δ*λ�*,*n*| for all *n* be Δ*λ�* :

$$|\Delta\lambda\_{\mathfrak{e},\mathcal{U}}| \le \Delta\lambda\_{\mathfrak{e}}.\tag{33}$$

By substituting Eqs. (32) and (33) into Eq. (28) and solving for Δ*λ�*, we obtain the condition

$$
\Delta\lambda\_{\varepsilon} < \frac{\bar{\lambda\_n}^2}{L} \left(\frac{1}{2(\alpha\_n' + 1)} - \varepsilon\right). \tag{34}
$$

For example, *α*� *n* = 2, *λ*¯ *n* = 800 nm, *L* =25 mm, and *�* = 1/12, Δ*λ�* should be less than 0.002 nm.

#### **5. Scale adjustment for phase calculation**

In the Fresnel-transform based method (FTM), the pixel size of the reconstructed image increases with the reconstruction distance and the recording wavenumber. The variation of the pixel size poses problems in holographic interferometry (Yamaguchi et al. (2006)). In contrast, the convolution method (CM) (Yamaguchi et al. (2002)) keeps the pixel size of the reconstructed image the same as the pixel size of an image sensor used for hologram recording. However, the CM requires huge computational time with respect to the FTM. A

and 785.14 nm were used for two-wavelength holographic interferometry. Figure 3 shows the phase differences extracted from the images reconstructed from the holograms using no scale adjustment. As shown in Fig. 3 (a), it is clear that the phase subtraction was correctly performed on the entire object with the wavelength difference of 0.20 nm. In contrast, the phase difference with the wavelength difference of 1.75 nm shown in Fig. 3 (b) has random structures in the right and the lower side. The random structures were caused by incorrect

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 61

(a) Δ*λ* = 0, 20 nm (b) Δ*λ* = 1.75 nm

The present method was applied to solve the problem in the phase subtraction for two-wavelength holographic interferometry. The bilinear interpolation (Kreis (2005)) was used for image magnification. Figure 4 shows phase difference distributions with a wavelength difference of 1.75 nm calculated using scale adjustment by application of the bilinear interpolation after and before reconstruction of the hologram. It is seen that a random structure appears on a part of the phase difference shown in Fig. 4 (a). As shown in Fig. 4 (b), it is clear that the phase subtraction was correctly performed on the entire object. This means that the present method completely solved the problem in the phase subtraction due to the

Figure 5 shows an optical setup for hologram recording. The light source was a laser diode (Hitachi HL7851G) with a wavelength of 785 nm and an output power of 50 mW. The image sensor was a CCD camera (Prosilica EC1350) having pixels of 1360 × 1024 with a pitch of 4.65

A beam emitted from the laser diode was collimated by a objective lens OL1 and then cleaned by a objective lens OL2 and a pinhole. The beam was again collimated by a lens L1 and separated into two beams by a half mirror. One was expanded by lenses L2 and L3 and reflected by a half mirror to illuminate the object. The angle of incidence *θ* was equal to zero.

Fig. 3. Phase difference distributions calculated using no scale adjustment.

changes in the pixel size.

**6. Experiment**

*μ*m × 4.65 *μ*m.

phase subtraction due to the changes in the pixel size of the reconstructed image.

zero padding method was proposed to control the pixel size of the image reconstructed by the FTM (ALFIERI et al. (2006); Ferraro et al. (2004)). In this method, an increase in the total pixel number of the hologram decreases the pixel size of the reconstructed image and can cause an explosive increase of the computational time as a result of the impossibility of the fast Fourier transform (FFT) application in the reconstruction. Algorithms based on an approach of splitting the reconstruction process into two diffraction processes were reported (Yu & Kim (2006); Zhang et al. (2004)). The methods require almost the twice computational time with respect to the FTM.

We present a simple and fast adjustment of pixel size in the reconstructed image. We adjust the pixel size by magnifying a recorded hologram before reconstruction of the hologram. The total pixel number of the hologram can be maintained for the possibility of the FFT application. Only one time calculation of the diffraction is required. Let consider that a hologram is recorded with the wavelength of *λ*1, and an another hologram is recorded with the wavelength of *λ*2. By application of a image magnification into the second hologram plane, the pixel sizes of the images reconstructed from the holograms are adjusted to be the same. Let the rescaled pixel size of the second hologram in a hologram plane be Δ*x*2 × Δ*y*2. For the adjustment, Δ*x*2 and Δ*y*2 should satisfy

$$
\Delta\_{x2} = \Delta\_x \frac{\lambda\_2}{\lambda\_1}, \Delta\_{y2} = \Delta\_y \frac{\lambda\_2}{\lambda\_1}. \tag{35}
$$

Experiments are performed to demonstrate the effectiveness of our pixel adjustment method. A test object was a Japanese one yen coin. Intensity and phase distributions of an image reconstructed from a hologram are shown in Fig. 2. As shown in Fig. 2 (b), the phase distribution has a random structure due to the light scattering from the diffusely surface of the object.

(a) Intensity distribution. (b) Phase distribution.

Fig. 2. Reconstructed image of hologram.

Holograms were recorded through changing the wavelength by injection current control of the laser diode. Three holograms with the recording wavelengths of 783.39 nm , 783.59 nm and 785.14 nm were used for two-wavelength holographic interferometry. Figure 3 shows the phase differences extracted from the images reconstructed from the holograms using no scale adjustment. As shown in Fig. 3 (a), it is clear that the phase subtraction was correctly performed on the entire object with the wavelength difference of 0.20 nm. In contrast, the phase difference with the wavelength difference of 1.75 nm shown in Fig. 3 (b) has random structures in the right and the lower side. The random structures were caused by incorrect phase subtraction due to the changes in the pixel size of the reconstructed image.

The present method was applied to solve the problem in the phase subtraction for two-wavelength holographic interferometry. The bilinear interpolation (Kreis (2005)) was used for image magnification. Figure 4 shows phase difference distributions with a wavelength difference of 1.75 nm calculated using scale adjustment by application of the bilinear interpolation after and before reconstruction of the hologram. It is seen that a random structure appears on a part of the phase difference shown in Fig. 4 (a). As shown in Fig. 4 (b), it is clear that the phase subtraction was correctly performed on the entire object. This means that the present method completely solved the problem in the phase subtraction due to the changes in the pixel size.

### **6. Experiment**

8 Hologram / Book 2

zero padding method was proposed to control the pixel size of the image reconstructed by the FTM (ALFIERI et al. (2006); Ferraro et al. (2004)). In this method, an increase in the total pixel number of the hologram decreases the pixel size of the reconstructed image and can cause an explosive increase of the computational time as a result of the impossibility of the fast Fourier transform (FFT) application in the reconstruction. Algorithms based on an approach of splitting the reconstruction process into two diffraction processes were reported (Yu & Kim (2006); Zhang et al. (2004)). The methods require almost the twice computational time with

We present a simple and fast adjustment of pixel size in the reconstructed image. We adjust the pixel size by magnifying a recorded hologram before reconstruction of the hologram. The total pixel number of the hologram can be maintained for the possibility of the FFT application. Only one time calculation of the diffraction is required. Let consider that a hologram is recorded with the wavelength of *λ*1, and an another hologram is recorded with the wavelength of *λ*2. By application of a image magnification into the second hologram plane, the pixel sizes of the images reconstructed from the holograms are adjusted to be the same. Let the rescaled pixel size of the second hologram in a hologram plane be Δ*x*2 × Δ*y*2.

Δ*x*2 = Δ*x*

*λ*2 *λ*1

Experiments are performed to demonstrate the effectiveness of our pixel adjustment method. A test object was a Japanese one yen coin. Intensity and phase distributions of an image reconstructed from a hologram are shown in Fig. 2. As shown in Fig. 2 (b), the phase distribution has a random structure due to the light scattering from the diffusely surface of

(a) Intensity distribution. (b) Phase distribution.

Holograms were recorded through changing the wavelength by injection current control of the laser diode. Three holograms with the recording wavelengths of 783.39 nm , 783.59 nm

, Δ*y*2 = Δ*y*

*λ*2 *λ*1

. (35)

respect to the FTM.

the object.

For the adjustment, Δ*x*2 and Δ*y*2 should satisfy

Fig. 2. Reconstructed image of hologram.

Figure 5 shows an optical setup for hologram recording. The light source was a laser diode (Hitachi HL7851G) with a wavelength of 785 nm and an output power of 50 mW. The image sensor was a CCD camera (Prosilica EC1350) having pixels of 1360 × 1024 with a pitch of 4.65 *μ*m × 4.65 *μ*m.

A beam emitted from the laser diode was collimated by a objective lens OL1 and then cleaned by a objective lens OL2 and a pinhole. The beam was again collimated by a lens L1 and separated into two beams by a half mirror. One was expanded by lenses L2 and L3 and reflected by a half mirror to illuminate the object. The angle of incidence *θ* was equal to zero.

(a) Top side of view. (b) Front side of view.

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 63

(a) The entire image. (b) Closeup in a 1st-order image.

The distance between the object and the camera was 300 mm. The field of view of a reconstructed object image was 50 mm × 50 mm. As shown in Fig.6, a test object was a

For keeping an aspect ratio of a reconstructed object image equal to 1.0, the central region with

The reconstructed image is shown in Fig. 7. The object light and its conjugate were reconstructed at the upper left and the lower right. Two bright points in the upper right

Holograms were recorded through changing the wavelength of a laser diode by injection current and operation temperature controls. The power spectral distribution of the laser light was measured using an optical spectrum analyzer(Ando Electric AQ-63515A) during hologram recording. The specification of the analyzer was as follows: the wavelength accuracy was 0.05 nm, the resolution was 0.05 nm, and the repeatability was 0.005 nm.

1024 × 1024 pixels of a hologram was used for the object image reconstruction.

and the lower left are produced by reflection at the back surface of a half mirror.

Fig. 6. Photograph of a test object.

Fig. 7. Image reconstructed from a hologram.

Japanese one yen coin.

Fig. 4. Phase difference distributions with a wavelength difference of 1.75 nm calculated using scale adjustment by application of the bilinear interpolation (a) after- and (b) beforereconstruction of the hologram.

Fig. 5. Hologram recording setup. OSA: optical spectrum analyzer; FC: fiber coupler; HM: half mirror; OL1: × 10 objective lens; OL2: 5 × objective lens; L1: lens (*f* =60 mm); L2: lens (*f*=70 mm); L3: lens (*f*=170 mm); L4: lens (*f*=60 mm).

The other beam was again splited by a half mirror. The light transmitted through the half mirror was coupled into a fiber coupler and detected by an optical spectrum analyzer. The light reflected by the half mirror was focused by a lens L4 and became a reference point light source.

(a) Top side of view. (b) Front side of view.

10 Hologram / Book 2

(a) (b)

L1 HM L2 L3

Fig. 5. Hologram recording setup. OSA: optical spectrum analyzer; FC: fiber coupler; HM: half mirror; OL1: × 10 objective lens; OL2: 5 × objective lens; L1: lens (*f* =60 mm); L2: lens

The other beam was again splited by a half mirror. The light transmitted through the half mirror was coupled into a fiber coupler and detected by an optical spectrum analyzer. The light reflected by the half mirror was focused by a lens L4 and became a reference point light

HM

3D Object

CCD Camera

HM

Fig. 4. Phase difference distributions with a wavelength difference of 1.75 nm calculated using scale adjustment by application of the bilinear interpolation (a) after- and (b) before-

HM L4

reconstruction of the hologram.

FC OSA

Laser Diode

source.

OL1 OL2

(*f*=70 mm); L3: lens (*f*=170 mm); L4: lens (*f*=60 mm).

(a) The entire image. (b) Closeup in a 1st-order image.

Fig. 7. Image reconstructed from a hologram.

The distance between the object and the camera was 300 mm. The field of view of a reconstructed object image was 50 mm × 50 mm. As shown in Fig.6, a test object was a Japanese one yen coin.

For keeping an aspect ratio of a reconstructed object image equal to 1.0, the central region with 1024 × 1024 pixels of a hologram was used for the object image reconstruction.

The reconstructed image is shown in Fig. 7. The object light and its conjugate were reconstructed at the upper left and the lower right. Two bright points in the upper right and the lower left are produced by reflection at the back surface of a half mirror.

Holograms were recorded through changing the wavelength of a laser diode by injection current and operation temperature controls. The power spectral distribution of the laser light was measured using an optical spectrum analyzer(Ando Electric AQ-63515A) during hologram recording. The specification of the analyzer was as follows: the wavelength accuracy was 0.05 nm, the resolution was 0.05 nm, and the repeatability was 0.005 nm.

calculated by Δ*φn* with Λ*n* are shown in Fig. 9. Figure 10 shows the plot of the object heights along a line denoted in white and black in Fig. 9 (i) as a function of the lateral positions. Figure

Multiple-Wavelength Holographic Interferometry with Tunable Laser Diodes 65

(a) Δ*φ*1 (b) Δ*φ*2 (c) Δ*φ*3

(d) Δ*φ*4 (e) Δ*φ*5 (f) Δ*φ*6

(g) Δ*φ*7 (h) Δ*φ*8 (i) Δ*φ*9

Height (mm)

Fig. 10. Plot of the object heights along a line denoted in (a) white and (b) black in Fig.9 (i) as

0 0.5 1 1.5 2 2.5 3

Position (mm) (b)

Fig. 9. An object height calculated from Δ*φn*

0 0.5 1 1.5 2 2.5 3

Position (mm)

(a)

a function of lateral positions.

Height (mm)

10 shows that the step-heights of 0.1 mm and 12 mm were correctly detected.

Fig. 8. Phase differences.

The central wavelength was obtained through the calculation of the center of gravity in the spectrum distribution measured using the optical spectrum analyzer.

Ten holograms for phase difference extraction were chosen from the holograms. The phase differences Ψ*n* were extracted from the object images reconstructed from the holograms with wavelengths *λn*. Figure 8 shows Ψ*n*. It can be seen in Fig. 8 that the phase differences have a salt-and-pepper noize produced by speckle noize. As shown in later, the noize can be suppressed by image processing method such as smoothing and median filtering (Yamaguchi (2001); Yamaguchi et al. (2006)).

The unwrapped phase differences Δ*φn* were retrieved by the recursive calculations of Eq. (19). The synthetic wavelengths were calibrated (Wada et al. (2008)) by the comparison between the phase differences and the object height measured by using a slide caliper. The calibrated synthetic wavelengths were Λ1 = 129.1mm, Λ2 = 34.02mm, Λ3 = 11.47mm, Λ4 = 8.593mm, Λ5 = 4.914mm, Λ6 = 2.849mm, Λ7 = 1.380mm, Λ8 = 0.9061mm, Λ9 = 0.4637mm.

The object profile was obtained using Δ*φn*. Since the incident angle *θ* for the object illumination was equal to zero, *L* = 2*h* and

$$h = \frac{\Lambda\_n \Delta \phi\_n}{4\pi}.\tag{36}$$

Because Δ*φ*1 = Ψ1 and a phase difference within (−*π*,*π*] corresponds to a object height within (−Λ*n*/4, Λ*n*/4], a measurable step height is Λ1/4 = 32 mm. The object height distributions calculated by Δ*φn* with Λ*n* are shown in Fig. 9. Figure 10 shows the plot of the object heights along a line denoted in white and black in Fig. 9 (i) as a function of the lateral positions. Figure 10 shows that the step-heights of 0.1 mm and 12 mm were correctly detected.

12 Hologram / Book 2

(a) Ψ1 (b) Ψ2 (c) Ψ3

(d) Ψ4 (e) Ψ5 (f) Ψ6

(g) Ψ7 (h) Ψ8 (i) Ψ9

The central wavelength was obtained through the calculation of the center of gravity in the

Ten holograms for phase difference extraction were chosen from the holograms. The phase differences Ψ*n* were extracted from the object images reconstructed from the holograms with wavelengths *λn*. Figure 8 shows Ψ*n*. It can be seen in Fig. 8 that the phase differences have a salt-and-pepper noize produced by speckle noize. As shown in later, the noize can be suppressed by image processing method such as smoothing and median filtering (Yamaguchi

The unwrapped phase differences Δ*φn* were retrieved by the recursive calculations of Eq. (19). The synthetic wavelengths were calibrated (Wada et al. (2008)) by the comparison between the phase differences and the object height measured by using a slide caliper. The calibrated synthetic wavelengths were Λ1 = 129.1mm, Λ2 = 34.02mm, Λ3 = 11.47mm, Λ4 = 8.593mm,

The object profile was obtained using Δ*φn*. Since the incident angle *θ* for the object

*h* = Λ*n*Δ*φn*

Because Δ*φ*1 = Ψ1 and a phase difference within (−*π*,*π*] corresponds to a object height within (−Λ*n*/4, Λ*n*/4], a measurable step height is Λ1/4 = 32 mm. The object height distributions

4*π* . (36)

Λ5 = 4.914mm, Λ6 = 2.849mm, Λ7 = 1.380mm, Λ8 = 0.9061mm, Λ9 = 0.4637mm.

spectrum distribution measured using the optical spectrum analyzer.

Fig. 8. Phase differences.

(2001); Yamaguchi et al. (2006)).

illumination was equal to zero, *L* = 2*h* and

(d) Δ*φ*4 (e) Δ*φ*5 (f) Δ*φ*6

(g) Δ*φ*7 (h) Δ*φ*8 (i) Δ*φ*9

Fig. 9. An object height calculated from Δ*φn*

Fig. 10. Plot of the object heights along a line denoted in (a) white and (b) black in Fig.9 (i) as a function of lateral positions.

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URL: *http://ao.osa.org/abstract.cfm?URI=ao-37-10-1917*

using tunable laser diodes, *Applied Optics* **47**(12): 2053. URL: *http://ao.osa.org/abstract.cfm?URI=ao-47-12-2053*

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URL: *http://ao.osa.org/abstract.cfm?id=44347*

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*Lasers in Engineering* **36**(5): 417–428.

Regularized Phase-Tracking System, *Applied Optics* **37**(10): 1917.

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Measurement, *Optics and Lasers in Engineering* **24**(2): 245–260.

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*Abstract&list\_uids=16463728*

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**45**(29): 7610–7616.


**1. Introduction**

Irritable bowel syndrome (IBS) is a common functional gastrointestinal disorder characterized by chronic recurrent abdominal pain, changes in bowel habits, and other symptoms such as bloating and flatulence. Based on the Rome IV criteria, four subtypes of IBS exist depending on the predominant stool pattern, including IBS with constipation (IBS-C), IBS with diarrhea (IBS-D), IBS with mixed bowel habits (IBS-M), and unclassified IBS [1,2]. IBS has a global prevalence of approximately 11% and is associated with several comorbidities, such as anxiety, depression, fibromyalgia, migraines, chronic pelvic pain, and others [3,4]. IBS is a major socioeconomic burden because affected patients utilize more healthcare resources with reduced work productivity when compared to the healthy population [5]. IBS is a complex heterogeneous condition with a multifactorial pathogenesis. Proposed mechanisms involved in the pathogenesis of IBS include visceral hypersensitivity, gut–brain axis alterations, disorders in the epithelial barrier integrity leading to abnormal mucosal intestinal permeability, changed intestinal motility, immune system activation, food intolerance, low-grade inflammation, altered enteroendocrine pathways

signaling, genetic basis (e.g., mutation in the SC5NA gene encoding a sodium channel ion; a number of single-nucleotide polymorphism studies have also identified polymorphisms in genes associated with IBS pathogenesis including genes coding for serotonin signaling, immune regulation, and epithelial barrier function), and the evolving concept of dysbiosis in the gut microbiota (Figure 1) [6–15].

**Figure 1.** Role of the microbiome in irritable bowel syndrome (IBS).

Gastrointestinal (GI) microbiota are the most relevant microbial community in the body and belong to the so-called microbiota which includes all microorganisms living in the human body. The intestinal microbiota includes bacteria, archaea, fungi, eukaryotes, and viruses. Most of the bacteria in the GI tract are represented in four main bacterial phyla: Firmicutes, Bacteroidetes, Proteobacteria, and Actinobacteria [16,17]. Trillions of microorganisms reside in the GI tract, with the highest density in the colon. However, most of them remain uncharacterized. Since this number is approximately equal to 10 times the total number of body cells, the interest in microbiota study for intestinal and extra-intestinal diseases is not surprising [18–22]. Advances in molecular biology techniques have significantly contributed to microbiome research. The most common technique to analyze the composition of the gut bacteria is marker gene sequencing, generally using the 16S rRNA gene [23–25]. Advantages of this method are related to its simplicity and low cost.

The aim of this review is to overview the most up-to-date literature about the evolving role of gut microbiome manipulation in IBS management, with a focus on probiotics, prebiotics, antibiotics, and fecal transplantation.


**5.3.2 The mode of transmission**

The hypothesis predicts a higher maternal transmission of schizophrenia. Although there has been no convincing evidence for maternal transmission of schizophrenia, several reports suggest a higher maternal transmission of schizophrenia (Shimizu et al., 1987; Goldstein et al., 1990; Valero et al., 1998; Li et al., 2007).

Some researchers have proposed the hypothesis that schizophrenia is associated with *de novo* mutations arising in paternal germ cells (Malaspina et al., 2001; Zammit et al., 2003; Byrne et al., 2003; El-Saadi et al., 2004; Sipos et al., 2004). It is based on the observation ('paternal age effect') that the risk of schizophrenia in the offspring seems to increase as the paternal age advances from 20 years to over 50 years.

However, the difference in the mean ages of fathers between affected and unaffected individuals are not very large (< 1.7 years) (Malaspina et al., 2001; El-Saadi et al., 2004). Furthermore, the risk of schizophrenia was also increased in the offspring of younger men

Impact of Epidemiology on Molecular Genetics of Schizophrenia 127

A role of oxidative stress in the pathogenesis of rheumatoid arthritis has been suggested by several lines of evidence (for review, see Hitchon et al, 2004). In addition, it has been shown that chronic oxidative stress in the synovial T lymphocytes is not secondary to exposure to environmental free radicals but originates from intracellularly produced reactive oxygen species (Remans et al, 2005). Therefore, a presumptive susceptibility gene for rheumatoid arthritis, which may cause intracellular oxidative stress in several cell lines, could be a facilitating gene for schizophrenia in this model and is likely to be subject to a negative

Indeed, robust evidence shows a negative association between schizophrenia and rheumatoid arthritis while the exact mechanism is still unknown (Vinogradov et al., 1991; Jablensky, 1995; Rubinstein, 1997; Oken 1999). According to the nuclear genome model, several hypotheses have been proposed such that pathogenic genes for schizophrenia may

Thus the mitochondrial genome model may offer a new explanation for the low comorbidity between schizophrenia and rheumatoid arthritis and the additional prediction: most of patients with both of the diseases would be females because the survival rate of males in early life stage must be remarkably reduced due to lack of the antioxidant defense by estrogen, and show more negative symptoms, poorer response to neuroleptic medication,

The hypothesis predicts that early-life exposure to environments which induce strong oxidative stress can increase the risk of later development of schizophrenia in the

Indeed, prenatal environmental factors such as severe nutritional deficiency (Susser, et al., 1996), exposure to increased homocysteine (Brown et al., 2007) or lead (Opler & Susser, 2005), and infection of influenza virus (Limosin et al., 2003; Brown et al., 2004; Opler & Susser, 2005) and Toxoplasma gondii (Brown et al., 2005) have been suggested to increase the risk for schizophrenia. More recently, it has been suggested that central nervous system infections of cytomegalovirus or mumps virus in childhood may also increase the risk for schizophrenia (Dalman et al., 2008). All of these factors have been shown to affect mitochondria, inducing strong intracellular oxidative stress and/or apoptosis (Akaike et al., 1990; Edlund et al., 1994; Speir et al., 1998; He et al., 2003; Berger et al., 2004; Zaki et al., 2005; Gupta et al., 2004; Kruman et al., 2006; Wang et al., 2006; Poncet et al., 2006; Chang et al.,

**5.3.6 Increased obstetric complications in the birth of patients with schizophrenia**

It has been suggested that mitochondrial dysfunction may be involved in the etiology of preeclampsia (Shanklin et al., 1990; Barton et al., 1991; Furui et al., 1994). In addition, a high incidence of preeclampsia, eclampsia, and stillborn infants has been observed in a family with a known mitochondrial disorder (Torbergsen et al. 1989). Folgero et al. (1996) demonstrated two separate mtDNA point mutations in two families having a high incidence

selection in the predisposed matrilineal pedigrees.

and/or more morphological changes in the brain.

**5.3.5 Prenatal risk factors for schizophrenia**

predisposed population.

2007; Nishikawa et al., 2007).

of preeclampsia and eclampsia.

be protective genes for rheumatoid arthritis and vice versa.

(< 21 years) (Malaspina et al., 2001; El-Saadi et al., 2004; Sipos et al., 2004) as well as in the offspring of younger women (< 20 years) (El-Saadi et al., 2004). Therefore, major roles of paternally derived mutations in schizophrenia seem to remain unsubstantiated.

Indeed, no available data can exclude the possibility that the 'paternal age effect' has a 'maternal origin'; while women in many countries today may be usually supposed to bear children after the age of 20 years and not to marry much older men or too young men unless the men have special socio-economic benefits, a certain proportion of predisposed women might behave differently.

It should be noted that in the famous twin study by Gottesman and Bertelsen (1989) which included almost equal number of male and female monozygotic twins, most schizophrenic twins whose offspring are affected are females (12 out of 14), implying that the transmission was mainly via females ( 14 14 14 14 12 14 13 14 14 *pC C C* 0.5 0.5 0.5 0.007 ). While this gender effect might be due to non-genetic factors such as stronger psychological interactions between mother and child, we must also consider the possibility that it is due to the closer genetic relationship between mother and child, i.e. the mtDNA.


**14. Summary**

are capable of producing different widths of deposits within a layer through varying travel speed and wire feed rate, while maintaining a constant deposit height. Therefore, we propose an adaptive path planning strategy that uses continuously varying step-over distances by adjusting the process parameters to deposit beads with variable width within any given path [28]. The developed adaptive MAT path planning algorithm is able to automatically generate path patterns with varying step-over distances (refer to **Figure 19e**) by analyzing geometry information to achieve better part quality (void-free deposition), accuracy at the boundary, and material efficiency, as shown in **Figure 19f**. Example of adaptive MAT path generation for a geometry with multiple holes is shown in **Figure 20**. Examples of void-free additive manu‐

**Figure 20.** Example of a geometry with multiple holes. (a) Computed branch loops as represented by red lines; (b) Div‐ ided six domains as represented by different colors; (c) Generated numerous radiations from the branch points and the

various decomposed simple shapes; (d) Generated adaptive paths for the geometry.

facturing using adaptive MAT paths could be found in [27].

20 New Trends in 3D Printing

This chapter presented slicing strategy for AM, namely unidirection slicing and multidirection slicing. Apart from the slicing, methods of existing path planning, including the newly published MAT-based non-adaptive and adaptive paths, were reviewed for advanced design in AM.

The main conclusions are listed as following: