**2. Digital holographic phase shifting interferometry**

In digital holography the recorded wavefield is reconstructed by multiplying the stored hologram values by the complex conjugate of a numerical model of the reference wave *r\*( ζ, η)* and then calculating the resulting diffraction field *b'( x', y')* in the image plane, see Fig. 1 (Kreis, 1996, 2005; Schnars & Jüptner, 2005). The numerical reconstruction of the digitally recorded hologram is defined as a numerical calculation of the phase and the intensity of the recorded wavefield. This is theoretically calculated by the diffraction integral;

$$b^\prime(\mathbf{x}^\prime, y^\prime) = \frac{1}{i\lambda} \left[ h \right]^\prime(\zeta^\prime \eta) r^\prime(\zeta^\prime \eta) \frac{\exp\{ik\rho\}}{\rho} \, d\zeta \, d\eta,\tag{1}$$

 with <sup>222</sup> *d x* ' ( ') ( ') *y* and *k* is the wave number (*k=2π/λ*).

Fig. 1. Geometry of digital Fresnel holography.

The finite discrete form of the Fresnel approximation to the diffraction integral is

$$\begin{split} b^\*(n\Delta\mathbf{x}^\prime, m\Delta\mathbf{y}^\prime) = A \sum\_{j=0}^{N-1} \sum\_{l=1}^{M-1} h(j\Delta\boldsymbol{\zeta}^\prime, l\Delta\boldsymbol{\eta}) r^\* \left( j\Delta\boldsymbol{\zeta}^\prime, l\Delta\boldsymbol{\eta} \right) \\ \times \exp\left\{ \frac{i\pi}{d^\prime\lambda} \left( j^2 \Delta\boldsymbol{\zeta}^2 + l^2 \Delta\boldsymbol{\eta}^2 \right) \right\} \exp\left\{ 2i\pi \left( \frac{jn}{N} + \frac{lm}{M} \right) \right\}. \end{split} \tag{2}$$

The parameters used in this formula for calculating the complex field in the image plane are given by the CCD array used, having *N × M* pixels of pixel pitches *Δζ* and *Δη* in the two orthogonal directions. The stored hologram is *h(jΔζ, lΔη)*. The distance between the object and the CCD is denoted by *d*, and normally *d'=d*. Complex factors not depending on the hologram under consideration are contained in a given specific CCD.

The pixel spacing in the reconstructed field is

72 Advanced Holography – Metrology and Imaging

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

In digital holography the recorded wavefield is reconstructed by multiplying the stored hologram values by the complex conjugate of a numerical model of the reference wave *r\*( ζ, η)* and then calculating the resulting diffraction field *b'( x', y')* in the image plane, see Fig. 1 (Kreis, 1996, 2005; Schnars & Jüptner, 2005). The numerical reconstruction of the digitally recorded hologram is defined as a numerical calculation of the phase and the intensity of the

<sup>1</sup> \* exp{ } '( ', ') ( , ) ( , ) , *ik bxy h r d d*

*h(ζ*,*η)*

 

   (1)

*η*

ζ *y*'

 

recorded wavefield. This is theoretically calculated by the diffraction integral;

*d x* ' ( ') ( ') *y* and *k* is the wave number (*k=2π/λ*).

The finite discrete form of the Fresnel approximation to the diffraction integral is

1 1

'( ', ') ( , )( , )

0

hologram under consideration are contained in a given specific CCD.

*j l b nx my A hj l r j l*

 

*N M*

*d d'*

Object-Plane Hologram-Plane

The parameters used in this formula for calculating the complex field in the image plane are given by the CCD array used, having *N × M* pixels of pixel pitches *Δζ* and *Δη* in the two orthogonal directions. The stored hologram is *h(jΔζ, lΔη)*. The distance between the object and the CCD is denoted by *d*, and normally *d'=d*. Complex factors not depending on the

2 22 2

\*

 

> 

(2)

*x'* 

*b'(x',y')* 

*z* 

Image-Plane

exp exp 2 . '

*<sup>i</sup> jn lm jl i <sup>d</sup> N M*

*i*

with <sup>222</sup>

*b(x,y)* 

x

Fig. 1. Geometry of digital Fresnel holography.

**2. Digital holographic phase shifting interferometry** 

parameters.

*y* 

$$
\Delta \mathbf{x'} = \frac{d' \lambda}{N \, \Delta \zeta} \qquad \qquad \text{and} \qquad \Delta y' = \frac{d' \lambda}{M \, \Delta \eta} \tag{3}
$$

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 theorem states that *b'* is given by

$$b^\prime = A^\prime F^{\cdot 1} \{ F\{h.r^\prime\} \cdot F\{\mathbf{g}\} \} \; \; \; \; \tag{4}$$

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) for this convolution approach is

$$
\Delta \mathfrak{x}' = \Delta \zeta' \qquad \text{and} \qquad \Delta y' = \Delta \eta \tag{5}
$$

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 holograms are given by

$$I\_n = a(\mathcal{L}, \eta) + b(\mathcal{L}, \eta) \cos(\phi(\mathcal{L}, \eta) + \phi\_{\text{Rn}}), \qquad n = 1, 2, 3, \dots \tag{6}$$

where *a( ζ, η)* and *b( ζ, η)* are the additive and the multiplicative distortions and *Rn* is the phase shift performed in the reference wave during recording of the holograms. In our case the phase shift is 90°, and it starts with 0*<sup>o</sup> Rn* . In this case we get a set of four linear equations that are point wise solved by a Gaussian least squares method(Kreis, 1996). The complex wavefield in the hologram plane can be calculated from

$$H(\zeta',\eta) = \left[I\_1(\zeta',\eta) \cdot I\_3(\zeta',\eta)\right] + i\left[I\_4(\zeta',\eta) \cdot I\_2(\zeta',\eta)\right] \tag{7}$$

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

$$I(\mathbf{x}', y') = \left| b'(\mathbf{x}', y') \right|^2 \,, \tag{8}$$

and the phase distribution is given by

$$\varphi(\mathbf{x'}, \mathbf{y'}) = \arctan\left\{ \frac{\text{Im}\left| b'(\mathbf{x'}, \mathbf{y'}) \right|}{\text{Re}\left| b'(\mathbf{x'}, \mathbf{y'}) \right|} \right\} \tag{9}$$

Then, the optical phase difference due to the used phase object such as GRIN optical waveguides can be extracted.

Digital Holographic Interferometric Characterization of Optical Waveguides 75

partially tunnel into the surrounding potential barrier before turning around. Only certain allowed energies (in the case of the particle), or propagation coefficients *β* (in the case of the

The squared value of the refractive index distribution at the core of the GRIN optical waveguide is shown in Fig. 2. At the centre of the core the refractive index has a maximum value *no* and then decreases to reach *nclad* at *x=±R* (*R* is the radius of the core). The squared

, ,

The guided wave problem requires only the knowledge of the square of the refractive index

*n2(x)* 

Fig. 2. The core of the GRIN optical fibre is confined between cladding media, arranged such that the squared refractive index *n2(x)* is larger than that of the surrounding media <sup>2</sup> *nclad* . In the TE case, the E field is polarized along the *y-* axis and propagating along the *z-* axis. The TE modes of the GRIN optical waveguide are obtained as the solutions of the following

*n2clad*

*-R R x*

22 2 <sup>2</sup> () 0 *<sup>y</sup> o y*

*y*

*o i E*

*o i E*

*w x* 

*w z* 

*x*

, (14)

, (15)

*H*

*z*

*H*

*kn x E*

 

*n x <sup>x</sup> n n RxR R*

*n R xR x*

,

, (12)

, (13)

optical waveguide) create a standing wave in one-dimensional system.

2

*clad*

*o*

<sup>2</sup> <sup>2</sup> <sup>2</sup>

**4.1 Symmetric optical waveguide** 

where *∆n = no - nclad*.

distribution.

value of the refractive index distribution is given by,

( )

equations (Conwell, 1973; Marcuse, 1973),

*<sup>y</sup>*

2

*d E*

*dx*
