**8. Acknowledgment**

86 Advanced Holography – Metrology and Imaging

(33)

*o n nn k* 

Eqs. (24), (25) and (26) can be used to calculate the effective indices *neff* of each mode for every part of the second zone. The first zone of the sample supports three modes. The first zone of the effective index *neff* for each mode is calculated. The numbers of modes in the first

Generally, the number of the modes and the values of the effective indices *neff* depend on

Fig. 13. Plots of the second mode *(q=1)* and sixth mode *(q=5)* calculated for GRIN optical

In this chapter the digital holographic phase shifting interferometric method is used to investigate and characterize GRIN optical waveguide samples. Phase shifting digital holography has the advantages of no dc term and no twin image in the reconstructed field as well as high accuracy due to the redundant data in the multiple holograms. The optical interference phase differences due to the variation of refractive indices of GRIN optical waveguides are extracted. A simple algorithm is used to recover the tilting of the extracted optical interference phase differences along the GRIN optical wavegide, to avoid the errors of the calculated mean values of optical phase differences due to the GRIN optical

waveguides. The broadening of optical waveguide radius or width is also reduced.

used to reconstruct the 3D refractive index profile of GRIN optical fibre.

indices *neff* associated with every mode are determined.

The advances of accurate reconstructed interference phase differences in combination of multilayer model are used to calculate the refractive index profiles of GRIN optical waveguide. Also, a simple algorithm based on the assumptions of the multilayer model is

The refractive index profiles of symmetric and asymmetric GRIN optical waveguides and their associated modes are calculated. The propagation coefficient *β* and hence the effective

*s s* <sup>3</sup>

and second parts are four and eight modes respectively.

two factors; the first one is the depth and the second one is *∆n*.

and

waveguide sample.

**7. Conclusions** 

The authors would like to express their gratitude to Professor A A Hamza for his fruitful discussions and comments. In addition, the authors are greatly indebted to the Unit of Research Management of Mansoura University for financial support. We are also acknowledging the encouragement and support of the Vice president of Mansoura University for Higher studies & Research."

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Automatic refractive index profiling of fibers by phase analysis method using


Yasuhiro Harada, Aizuddin Wan and Hiroyasu Sone

In digital holography (Kreis, 2005; Schnars & Jueptner, 2005), holograms are recorded using an image sensor such as CCD cameras, and reconstruction of images of the object is performed by a computer. From this, digital holography has following attractive features: A wet process for developing recording media is not required; qualitative evaluation is easy for three-dimensional images of objects; and focused images of three-dimensional objects at a desired depth can be instantaneously acquired without a mechanical focusing process. However, this technology suffers from lower resolution of image sensors than photographic materials, and therefore, object size to be recorded is restricted to be large and far from the image sensor in off-axis recording geometry. These limitations can be moderated by introducing the phase-shifting method (Schreiber et al., 2007) that enables us to derive complex amplitude of the object wave on the image sensor directly from three or more phase-shifted holograms recorded in the in-line geometry (Yamaguchi & Zhang, 1997). However, since this method requires to record at least three or four phase-shifted hologram,

**Single-Shot Phase-Shifting Digital Holography** 

**Based on the Spatial Carrier Interferometry and** 

In order to resolve this problem, various single-shot technologies with phase-shifting features in subsequent holograms are proposed(Awatsuji et al., 2004; Nomura et al., 2006; Toge et al., 2008; Wyant, 2003). All of these methods are based on space-division multiplexing of the phase-shifted holograms in the single recording but device and technique used are different each other. In the former three methods, special micro-element arrays of phase retarders (Awatsuji et al., 2004) or polarizers (Nomura et al., 2006; Wyant, 2003) suitable to a CCD array for recording are strictly required, and therefore, any satisfactory result for practical instrumentations has not been reported until now. On the other hand, the last method only require an introduction of off-axis plane wave illumination to an image sensor for recording single holograms (Toge et al., 2008). From its simplicity, this method seems to be one of most practical candidate for single-shot digital holography. However, it should be point out that we must align the incident angle of the reference wave plane wave so that the relative phase difference between adjacent pixels of the image sensor becomes 2*π*/3 or *π*/2 radian. This means that very rigorous alignment in reference wave illumination is required. Furthermore, this requirement may be more serious problem in the case of color digital holography where multicolor laser light is utilized. In this situation, optical alignment should be independently

it is useless for instantaneous measurement of the dynamic object.

**1. Introduction**

*Kitami Institute of Technology*

**Its Tolerance Analysis** 

*Japan*

**5**

