**1. Introduction**

16 Hologram / Book 2

68 Advanced Holography – Metrology and Imaging

Yamaguchi, I., Matsumura, T. & Kato, J.-i. (2002). Phase-shifting color digital holography,

Yonemura, M. (1985). Wavelength-change characteristics of semiconductor lasers and their

Yu, L. & Kim, M. K. (2006). Pixel resolution control in numerical reconstruction of digital

Zhang, F., Yamaguchi, I. & Yaroslavsky, L. P. (2004). Algorithm for reconstruction of digital holograms with adjustable magnification, *Optics Letters* **29**(14): 1668.

*Surface+shape+measurement+by+phase-shifting+digital+holography*

application to holographic contouring, *Optics L etters* **10**: 1–3.

URL: *http://ol.osa.org/abstract.cfm?URI=ol-27-13-1108*

URL: *http://ol.osa.org/abstract.cfm?URI=ol-31-7-897*

URL: *http://ol.osa.org/abstract.cfm?URI=ol-29-14-1668*

URL: *http://ol.osa.org/abstract.cfm?id=8352*

holography, *Optics Letters* **31**(7): 897.

*Optics Letters* **27**(13): 1108.

URL: *http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?dispmax=50&DB=pubmed&term=*

Holography was developed by Dennis Gabor 1947. Gabor presented holography as a lensless process for image formation by reconstructed wavefronts (Gabor, 1948, 1949, 1951). Holography can be defined as a method for recording and reconstructing whole optical waveelds, which means intensity and phase (Gabor, 1948, 1949), thus it exhibits 3D characteristics like depth of field or parallax. Holographic interferometry (Powell & Stetson, 1965) is a very effective non-destructive, contactless tool to measure shape, deformation or refractive index distributions (Kreis, 1996).

In 1994 the modern digital holography was introduced (Schnars, 1994; Schnars, & Jüptner, 1994a; Schnars, & Jüptner, 1994b; Schnars et al., 1995; Schnars, & Jüptner, 2005). The digital holography can be defined as digital recording of the holograms and the numerical reconstruction of the wave fields in a computer, where, the charge coupled devices (CCDs) are the most frequently used devices to record the holograms. The digital holography in the last ten years was involved in a lot of applications due to the development of powerful computers, ultra large memories and smaller pixel size CCD targets (Kreis, 2005).

The phase shifting interferometric (PSI) technique was introduced by Hariharan et al. into the field of holography as an accurate method for real time fringe measurement (Hariharan et al., 1982). Furthermore, PSI combined with digital holography (Skarman et al., 1996). Yamaguchi and Zhang 1997 improved phase shifting digital holography (PSDH) (Yamaguchi & Zhang, 1997). The phase difference using PSDH is measured with an accuracy of 2/200 (Hariharan, 2002).

For a long time, the determination of refractive index distributions in fibres, optical waveguides or other transparent solids was performed by interferometric methods. Twobeam and multiple beam based interferometers were used as a non-destructive tool to determine the optical parameters of fibres (Faust, 1952, 1954; Marhic et al., 1975). While, the mathematics used in (Marhic et al., 1975; Saunder & Gardner, 1977; Barakat et al., 1985) neglect the non-straightforward refraction of the light beam inside the fibres. Hamza et al. (Hamza et al., 1994, 1995) constructed an accurate mathematical model (multilayer model) which considered the exact local refraction of the incident beam on its way through the graded index optical fibre, which is divided into a large number of thin concentric layers of constant refractive index. This model was verified with two-beam and multiple-beam interferometers. The consideration of incident beam refraction gave a better accuracy in the

Digital Holographic Interferometric Characterization of Optical Waveguides 71

The UV-laser lithographic fabrication of integrated optical waveguides has been wellrecognized in the last years. The UV excimer laser has been used to irradiate homogeneous slabs of PMMA in order to modify the refractive index locally in a controllable way at the polymeric substrate surface (Wochnowski et al., 2000; Shams-Eldin et al., 2004, 2005). Thus the integrated optical waveguides can be directly laser-written into the surface of polymeric substrates by employing lithographic masks. It is very important that the refractive index is modified in a controllable way by using specific conditions during the irradiation process in order to obtain the desired waveguide structures (Wochnowski et al., 2000). The UV photons of the incident laser radiation do interact with the molecules of the polymeric material due to a UV-photon-induced reaction (Shams-Eldin et al., 2005), so the refractive index at the surface of the PMMA slabs was changed in the irradiated area. Consequently, the refractive index modification in the irradiated area appears as a smooth variation, which

The mode field distribution across the GRIN optical waveguides is very important in applications. In the case of GRIN optical waveguides, the variation in refractive index can be presented as a function of the form of a complementary error function, exponential, Gaussian, parabolic, or shape not described by any simple mathematical function (Chiang, 1985; Mathey & Jullien, 1996). To use such optical waveguides in telecommunication, it is useful to know the number of propagated modes and their propagation constants for any given profile. The number of modes depends on two factors; the first one is the difference between maximum and minimum refractive indices of the guiding region, the second one is the radius of the guiding region. Eguchi and Horinouchi (Eguchi & Horinouchi, 2004) used the finite-element method to determine the number of modes of optical fibres. Shemirani et al. (Shemirani et al., 2009) developed a field-coupling model for propagation in a graded index optical fibre, analogous to the principal states model for polarization mode dispersion in the single mode fibre. This model was based on the concept of the first order principal modes, which have well defined group delays that depend on the strength of the mode coupling. This first-order model predicts a linear relationship between the intensity distributions at the input and output. This model is extended to account for higher order

In this Chapter, digital holographic phase shifting interferometry is used to investigate the optical properties of graded index optical waveguides, e.g. graded index planar optical waveguides and optical fibres. A planar optical waveguide sample is prepared by the UV laser lithographic method. The approach of digital holography with the aid of phase shifting interferometry is applied. The reconstructed optical phase differences along the GRIN optical waveguides are extracted and then a simple algorithm is used to avoid the problem of tilted GRIN optical waveguides inside the optical field. The extracted optical phase differences due to the optical waveguides are aligned to be perpendicular to the *x*-axis. After this process, the mean optical phase differences across the samples are calculated and thus the errors in the calculated mean values of optical phase differences across the optical waveguides are reduced. The proposed method is used to construct the refractive index profiles across GRIN optical waveguides with the aid of the multilayer model. Also, a simple algorithm is used to reconstruct the 3D refractive index of the GRIN optical fibre considering the symmetrical distribution of the GRIN optical fibre layers. In addition, an analytical method is presented to calculate the effective indices and the mode field

depends on the waveguide depth.

modal dispersion (Shemirani & Kahn, 2009).

determination of the optical parameters of graded index optical fibres (Hamza et al., 1995, 2001), thick optical fibres (Hamza et al., 2004) and mechanically stressed optical fibres (Sokkar et al., 2008) than without this consideration. Automated Fizeau interferometric techniques were applied in studies of the optical and optothermal properties of fibres (El-Morsy et al., 2002a, 2002b; Hamza et al., 2007). These cited developments substantially increased the accuracy of the measured optical fibre parameters.

In addition, digital holographic interferometry (DHI) is used for determining refractive index profiles. Digital holographic microscopy (DHM) was used to measure the mean integral refractive index and thickness of living cells (Rappaz et al., 2005). The absolute accuracy of the mean refractive index measurement was about 0.0003. Also, Kemper et al. (Kemper & Carl, 2006; Kemper et al., 2007) used DHI to measure the refractive index and thickness of living cells. Kebbel et al. applied digital holography to refractive index variations within transparent media in microgravity experiments (Kebbel et al., 1999). Two-dimensional refractive index profiles of phase gratings have been investigated using DHI (De Angelis et al., 2006) as well as the refractive indices of liquids using lensless Fourier DHI (Hossain et al., 2006). The high performance of DHI was tested by measurements of low variation refractive indices of fluids in a comparative study with other techniques (Dubois et al., 1999; Owen & Zozulya, 2002) like traditional Mach-Zehnder interferometry. The coupling of digital holographic microscopy and polarization imaging digital holography was demonstrated in an investigation of induced birefringence in non striped bent optical fibres and the birefringence of stressed PMMA (Cuche et al., 1999; Colomb et al., 2002, 2005). The mathematics describing the refractive index of transparent materials used in (Dubois et al., 1999; Owen & Zozulya, 2002; De Angelis et al., 2006; Hossain et al., 2006) cannot be used directly to measure and configure the refractive index profile of GRIN optical waveguides, since it assumes a constant refractive index along the light path in a material. So it can be used only to determine the mean refractive index of the GRIN optical waveguides but it does not consider the varying refraction of the beam along its path inside the fibre. Large scale strongly refracting fields produce ray bending. This effect was recognized in holographic interferometric investigations combined with iterative calculations (Sweeney & Vest, 1973) as well as tomographic methods (Cha & Vest, 1981).

Recently, digital holographic phase shifting interferometry with the aid of mathematical models, which consider the refraction of the incident rays, were used to investigate some optical parameters of fibrous materials (Wahba & Kreis, 2009a, 2009b, 2009c; Yassien et al., 2010). The refractive index profile and optical parameters of graded index fibres as well as the refractive index profile of bent optical fibres were determined with accuracy 2.3x10-4.

Organic or inorganic optical waveguides are made of transparent dielectric materials, e.g. graded index (GRIN) planar optical waveguides and GRIN optical fibres. The fabrication of integrated optics is a very crucial technology for optical communications and sensing devices in the near future. However, it is absolutely necessary to have accurate knowledge about the optical waveguide parameters to use it in technological applications.

In recent times, polymeric integrated optics has shifted to the focus of interest due to the low material cost and simple handing and processing of polymers during the manufacturing process. Polymethyl methacrylate (PMMA) is a very promising material to be used as a basic material to produce optical waveguides and gratings (Baker & Dyer, 1993; Eldada & Shacklette, 2000; Vollertsen & Wochnowski, 2004).

determination of the optical parameters of graded index optical fibres (Hamza et al., 1995, 2001), thick optical fibres (Hamza et al., 2004) and mechanically stressed optical fibres (Sokkar et al., 2008) than without this consideration. Automated Fizeau interferometric techniques were applied in studies of the optical and optothermal properties of fibres (El-Morsy et al., 2002a, 2002b; Hamza et al., 2007). These cited developments substantially

In addition, digital holographic interferometry (DHI) is used for determining refractive index profiles. Digital holographic microscopy (DHM) was used to measure the mean integral refractive index and thickness of living cells (Rappaz et al., 2005). The absolute accuracy of the mean refractive index measurement was about 0.0003. Also, Kemper et al. (Kemper & Carl, 2006; Kemper et al., 2007) used DHI to measure the refractive index and thickness of living cells. Kebbel et al. applied digital holography to refractive index variations within transparent media in microgravity experiments (Kebbel et al., 1999). Two-dimensional refractive index profiles of phase gratings have been investigated using DHI (De Angelis et al., 2006) as well as the refractive indices of liquids using lensless Fourier DHI (Hossain et al., 2006). The high performance of DHI was tested by measurements of low variation refractive indices of fluids in a comparative study with other techniques (Dubois et al., 1999; Owen & Zozulya, 2002) like traditional Mach-Zehnder interferometry. The coupling of digital holographic microscopy and polarization imaging digital holography was demonstrated in an investigation of induced birefringence in non striped bent optical fibres and the birefringence of stressed PMMA (Cuche et al., 1999; Colomb et al., 2002, 2005). The mathematics describing the refractive index of transparent materials used in (Dubois et al., 1999; Owen & Zozulya, 2002; De Angelis et al., 2006; Hossain et al., 2006) cannot be used directly to measure and configure the refractive index profile of GRIN optical waveguides, since it assumes a constant refractive index along the light path in a material. So it can be used only to determine the mean refractive index of the GRIN optical waveguides but it does not consider the varying refraction of the beam along its path inside the fibre. Large scale strongly refracting fields produce ray bending. This effect was recognized in holographic interferometric investigations combined with iterative calculations (Sweeney & Vest,

Recently, digital holographic phase shifting interferometry with the aid of mathematical models, which consider the refraction of the incident rays, were used to investigate some optical parameters of fibrous materials (Wahba & Kreis, 2009a, 2009b, 2009c; Yassien et al., 2010). The refractive index profile and optical parameters of graded index fibres as well as the refractive index profile of bent optical fibres were determined with accuracy

Organic or inorganic optical waveguides are made of transparent dielectric materials, e.g. graded index (GRIN) planar optical waveguides and GRIN optical fibres. The fabrication of integrated optics is a very crucial technology for optical communications and sensing devices in the near future. However, it is absolutely necessary to have accurate knowledge

In recent times, polymeric integrated optics has shifted to the focus of interest due to the low material cost and simple handing and processing of polymers during the manufacturing process. Polymethyl methacrylate (PMMA) is a very promising material to be used as a basic material to produce optical waveguides and gratings (Baker & Dyer, 1993; Eldada &

about the optical waveguide parameters to use it in technological applications.

increased the accuracy of the measured optical fibre parameters.

1973) as well as tomographic methods (Cha & Vest, 1981).

Shacklette, 2000; Vollertsen & Wochnowski, 2004).

2.3x10-4.

The UV-laser lithographic fabrication of integrated optical waveguides has been wellrecognized in the last years. The UV excimer laser has been used to irradiate homogeneous slabs of PMMA in order to modify the refractive index locally in a controllable way at the polymeric substrate surface (Wochnowski et al., 2000; Shams-Eldin et al., 2004, 2005). Thus the integrated optical waveguides can be directly laser-written into the surface of polymeric substrates by employing lithographic masks. It is very important that the refractive index is modified in a controllable way by using specific conditions during the irradiation process in order to obtain the desired waveguide structures (Wochnowski et al., 2000). The UV photons of the incident laser radiation do interact with the molecules of the polymeric material due to a UV-photon-induced reaction (Shams-Eldin et al., 2005), so the refractive index at the surface of the PMMA slabs was changed in the irradiated area. Consequently, the refractive index modification in the irradiated area appears as a smooth variation, which depends on the waveguide depth.

The mode field distribution across the GRIN optical waveguides is very important in applications. In the case of GRIN optical waveguides, the variation in refractive index can be presented as a function of the form of a complementary error function, exponential, Gaussian, parabolic, or shape not described by any simple mathematical function (Chiang, 1985; Mathey & Jullien, 1996). To use such optical waveguides in telecommunication, it is useful to know the number of propagated modes and their propagation constants for any given profile. The number of modes depends on two factors; the first one is the difference between maximum and minimum refractive indices of the guiding region, the second one is the radius of the guiding region. Eguchi and Horinouchi (Eguchi & Horinouchi, 2004) used the finite-element method to determine the number of modes of optical fibres. Shemirani et al. (Shemirani et al., 2009) developed a field-coupling model for propagation in a graded index optical fibre, analogous to the principal states model for polarization mode dispersion in the single mode fibre. This model was based on the concept of the first order principal modes, which have well defined group delays that depend on the strength of the mode coupling. This first-order model predicts a linear relationship between the intensity distributions at the input and output. This model is extended to account for higher order modal dispersion (Shemirani & Kahn, 2009).

In this Chapter, digital holographic phase shifting interferometry is used to investigate the optical properties of graded index optical waveguides, e.g. graded index planar optical waveguides and optical fibres. A planar optical waveguide sample is prepared by the UV laser lithographic method. The approach of digital holography with the aid of phase shifting interferometry is applied. The reconstructed optical phase differences along the GRIN optical waveguides are extracted and then a simple algorithm is used to avoid the problem of tilted GRIN optical waveguides inside the optical field. The extracted optical phase differences due to the optical waveguides are aligned to be perpendicular to the *x*-axis. After this process, the mean optical phase differences across the samples are calculated and thus the errors in the calculated mean values of optical phase differences across the optical waveguides are reduced. The proposed method is used to construct the refractive index profiles across GRIN optical waveguides with the aid of the multilayer model. Also, a simple algorithm is used to reconstruct the 3D refractive index of the GRIN optical fibre considering the symmetrical distribution of the GRIN optical fibre layers. In addition, an analytical method is presented to calculate the effective indices and the mode field

Digital Holographic Interferometric Characterization of Optical Waveguides 73

' ' ' ' *d d <sup>x</sup> and y N M*

An alternative to the Fresnel approximation uses the fact that Eq. (1) describes a convolution of *h(ζ, η)r\*( ζ, η)*, with the impulse response *g(x', y', ζ, η)=(exp{ikρ})/iλρ*. The convolution

where *F* denotes the Fourier transform and *F-1* is its inverse. In practice, both *F* and *F-1* are calculated by the fast Fourier transform algorithm. The resulting pixel spacing (Kreis, 2005)

*x and y* ' '

The use of a real hologram in the Fresnel reconstruction or the convolution reconstruction, leads to a strong d.c. term, a focused real image, and a virtual image that is not sharp. The complex field can be recorded and calculated by phase-shifting digital holography. The calculated complex wavefield is used instead of a real hologram in the convolution approach to overcome the problems of the d.c. term and twin image. For this purpose several holograms, at least three, with known mutual phase shifts are recorded. These

( , ) ( , )cos( ( , ) ), 1,2,3,..... *<sup>n</sup> Rn Ia b*

phase shift performed in the reference wave during recording of the holograms. In our case

equations that are point wise solved by a Gaussian least squares method(Kreis, 1996). The

*H I I iI I* ( , ) ( , ) ( , ) ( , ) ( , ).

Finally, the reconstruction process is based on the use of the complex wavefield in the convolution algorithm. The intensity distribution in the reconstruction plane is given by

<sup>2</sup> *Ix y b x y* ( ', ') '( ', ') , (8)

Im '( ', ') ( ', ') arctan Re '( ', ') *bxy x y bxy*

Then, the optical phase difference due to the used phase object such as GRIN optical

   

> 

 

where *a( ζ, η)* and *b( ζ, η)* are the additive and the multiplicative distortions and


(5)

*n* (6)

*Rn* . In this case we get a set of four linear

 

(9)

1 3 42 - - (7)

*Rn* is the

(3)

The pixel spacing in the reconstructed field is

theorem states that *b'* is given by

for this convolution approach is

holograms are given by

the phase shift is 90°, and it starts with 0*<sup>o</sup>*

 

and the phase distribution is given by

waveguides can be extracted.

 

complex wavefield in the hologram plane can be calculated from

 

distribution across the symmetric and asymmetric GRIN optical waveguides. The digital holographic phase shifting interferometric approach affects the accuracy of the calculated parameters.
