**3. Refractive index profile of GRIN optical fibre using the multilayer model**

The GRIN optical fibre sample can be considered as a phase object. When it is implemented in the path of one of the interfering beams of a Mach-Zehnder interferometer, the produced fringes suffer shifts in the GRIN optical fibre region. These shifts represent the optical path difference between the light beam traversing the fibre and that traversing the surrounding medium.

In GRIN optical fibre, the core is surrounded by a homogeneous cladding of refractive index *ncl*. In the multilayer model, the core and cladding of the graded index optical fibre are assumed to consist of a large number of thin layers. Each layer has an annular cross section and is considered to have a thickness *a* and refractive index *nj* where the layers are numbered by *j =1,...,N* with *N = R/a* . *R* is the core radius. If this optical fibre is immersed in a liquid of refractive index *nL*, there will be a refraction of the incident beam at the boundary between the liquid and the cladding. Hamza et al. obtained a recurrence relation which predicts the optical path of the refracted beam through *Q* layers of the fibre. The GRIN fibre is illuminated by a collimated beam, with a ray crossing the centre of the core dening the optical axis (Hamza et al., 1995). An arbitrary beam transverses the fibre at a distance *dQ* from the optical axis and leaves the fibre at a distance *xQ*. Assume a coordinate system whose origin is at the fibre centre, then the corresponding fringe shift is defined as *ZQ* and the optical path difference *δQ* is given by the recurrence formula

$$\begin{split} \frac{\lambda Z\_{\mathcal{Q}}}{h} &= \frac{\lambda}{2\pi} \, \Delta \phi\_{\mathcal{Q}} = \sum\_{j=1}^{\mathcal{Q}} 2n\_{j} \Big( \sqrt{\left(\mathcal{R} \cdot (j \cdot 1) a\right)^{2} \cdot d\_{\mathcal{Q}}^{2} n\_{L}^{2} \left/ \ n\_{j}^{2}} \cdot \sqrt{\left(\mathcal{R} \cdot j a\right)^{2} \cdot d\_{\mathcal{Q}}^{2} n\_{L}^{2} \left/ \ n\_{j}^{2}} \right) \Big) \\ &+ 2n\_{\mathcal{Q}} \Big( \sqrt{\left(\mathcal{R} \cdot (\mathcal{Q} \cdot 1) a\right)^{2} \cdot d\_{\mathcal{Q}}^{2} n\_{L}^{2} \left/ \ n\_{Q}^{2}} \right) \cdot n\_{L} \Big( \sqrt{\mathcal{R}^{2} \cdot d\_{\mathcal{Q}}^{2}} + \sqrt{\mathcal{R}^{2} \cdot x\_{Q}^{2}} \Big) \end{split} \tag{10}$$

with *Q* running from 1 to *N* , *h* is the interfringe spacing and *ΔφQ* is the interference phase difference.

This recurrence relation also describes the shape of the fringes which is produced as a result of the refraction of the incident beam through the *Q* layers of the fibre. Without loss of generality, we can assume that the incident light beam passes through the middle of the *Q*th layer. Thus the value of *dQ* in Eq. (10) can be obtained from the following relation

$$d\_Q = \frac{\eta [R - (Q - 0.5)a]}{n\_L} \tag{11}$$

Eq. (10) can now be used to get the refractive index profile for GRIN optical fibres (*N*→∞). Hamza et al. have shown that inhomogeneous refraction must be considered in the measurement and calculation of optical fibre parameters, so that the accuracy of the measured parameters of GRIN optical fibre is increased (Hamza et al., 1994, 1995).

### **4. Characterization of TE modes in the GRIN optical waveguides**

The study of the GRIN optical waveguides using the wave equation to achieve the allowed propagating modes is familiar from basic quantum mechanics. The analogy between the particle in box and optical waveguide problem is very strong: both situations describe waves, which are confined between two reflecting boundaries. In both cases, the waves 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 optical waveguide) create a standing wave in one-dimensional system.

### **4.1 Symmetric optical waveguide**

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 value of the refractive index distribution is given by,

$$m^2(\mathbf{x}) = \begin{cases} m\_{\text{dual}}^2 & -R > \mathbf{x} \text{, } R < \mathbf{x} \\\\ n\_o - \Delta n \left(\frac{\mathbf{x}}{R}\right)^2 & -R < \mathbf{x} > R \end{cases} \tag{12}$$

where *∆n = no - nclad*.

74 Advanced Holography – Metrology and Imaging

The GRIN optical fibre sample can be considered as a phase object. When it is implemented in the path of one of the interfering beams of a Mach-Zehnder interferometer, the produced fringes suffer shifts in the GRIN optical fibre region. These shifts represent the optical path difference between the light beam traversing the fibre and that traversing the surrounding

In GRIN optical fibre, the core is surrounded by a homogeneous cladding of refractive index *ncl*. In the multilayer model, the core and cladding of the graded index optical fibre are assumed to consist of a large number of thin layers. Each layer has an annular cross section and is considered to have a thickness *a* and refractive index *nj* where the layers are numbered by *j =1,...,N* with *N = R/a* . *R* is the core radius. If this optical fibre is immersed in a liquid of refractive index *nL*, there will be a refraction of the incident beam at the boundary between the liquid and the cladding. Hamza et al. obtained a recurrence relation which predicts the optical path of the refracted beam through *Q* layers of the fibre. The GRIN fibre is illuminated by a collimated beam, with a ray crossing the centre of the core dening the optical axis (Hamza et al., 1995). An arbitrary beam transverses the fibre at a distance *dQ* from the optical axis and leaves the fibre at a distance *xQ*. Assume a coordinate system whose origin is at the fibre centre, then the corresponding fringe shift is defined as *ZQ* and the optical path difference *δQ* is given by the recurrence

*Q j Q L j Q L j*


*n R j a dn n R ja dn n*

2 22 2 2 2 2 2

2 2 22 2 22 2

(11)

, (10)


*Q QL Q L Q Q*

with *Q* running from 1 to *N* , *h* is the interfringe spacing and *ΔφQ* is the interference phase

This recurrence relation also describes the shape of the fringes which is produced as a result of the refraction of the incident beam through the *Q* layers of the fibre. Without loss of generality, we can assume that the incident light beam passes through the middle of the *Q*th

[ ( 0.5) ]

*L nR Q a <sup>d</sup> n*

Eq. (10) can now be used to get the refractive index profile for GRIN optical fibres (*N*→∞). Hamza et al. have shown that inhomogeneous refraction must be considered in the measurement and calculation of optical fibre parameters, so that the accuracy of the

The study of the GRIN optical waveguides using the wave equation to achieve the allowed propagating modes is familiar from basic quantum mechanics. The analogy between the particle in box and optical waveguide problem is very strong: both situations describe waves, which are confined between two reflecting boundaries. In both cases, the waves

*n R Q a dn n n R d R x*

layer. Thus the value of *dQ* in Eq. (10) can be obtained from the following relation

measured parameters of GRIN optical fibre is increased (Hamza et al., 1994, 1995).

**4. Characterization of TE modes in the GRIN optical waveguides** 

*Q*

2 ( 1) / / <sup>2</sup>

1

*Q*

*j*

1

2 ( 1) /

**3. Refractive index profile of GRIN optical fibre using the multilayer model** 

medium.

formula

difference.

*Q*

*Z*

*h*

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

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 equations (Conwell, 1973; Marcuse, 1973),

$$\frac{d^2 E\_y}{dx^2} + \left(k\_o^2 \, n^2(\mathbf{x}) - \beta^2\right) E\_y = 0 \qquad , \tag{13}$$

$$-\frac{i}{w\mu\_o}\frac{\partial E\_y}{\partial z} = H\_x$$

$$\epsilon\_r \tag{14}$$

$$\frac{\text{i}}{w\mu\_o} \frac{\partial E\_y}{\partial \mathbf{x}} = H\_z \tag{15}$$

Digital Holographic Interferometric Characterization of Optical Waveguides 77

On the other hand, to find the allowed TE modes and the propagation coefficients of the asymmetric GRIN optical waveguide, we assume that, the GRIN optical waveguide, as shown in Fig. 3, consists of two graded index zones; both have a Gaussian refractive index profile. The first zone is in the range *0 < x < b1*, and the second one is in the range *b2 < x < b4*. There is a very thin layer of constant refractive index *ns1* confined between the two zones in the range *b1 < x < b2*. The plane at *x=0* represents the interface between the surface of the first zone of the optical waveguide and the immersion liquid (of refractive index *nL*) and the plane at *x = b4* represents the interface between the second zone of the waveguide and the

The refractive index profile for each zone is fitted using a Gaussian function which could be used to determine the allowed TE modes for the optical waveguide sample. The first zone of the waveguide in the range *0 < x < b1* is represented by the Gaussian refractive index profile given by

> 2 <sup>1</sup> 1 1 ( ) , 0 *x <sup>b</sup> nx n n e <sup>s</sup> x b*

where *b* is the depth of the first zone (between *0* and *b1*), and *n1* is the difference between the refractive indices at *x = 0* and *x = b1*. The conditions for the wave propagation along the first zone of the waveguide are: (1) a zigzag of the beam path and (2) the total phase change must be a multiple of 2*π*. Using WKB approximation (Marcuse, 1973; Mathey, 1996), the

12 13

*<sup>o</sup> k n x dx*

where *β = koneff* is the propagation constant of the *m*th mode; *m* is an integer. *M* is the total number of modes, *ko = 2π/λ* is the wave-vector, *λ* is the wavelength, and *neff* is the effective

**Seconed zone**

*xc*

*<sup>b</sup> b3 <sup>4</sup> b1 b2*

Fig. 3. The waveguide confined between liquid and substrate media, arranged such that the

refractive index *n(x)* is larger than that of the surrounding media (*nL* ) and (*ns*).

**Depth (***b***) m**

2 () 2 2 2 , 1

(22)

 

*m mM* (23)

*ns*

*x*

**Substrat region**

**4.2 Asymmetric optical waveguide** 

substrate, which has refractive index *ns*.

**4.2.1 TE modes in the first zone of waveguide** 

1

*b*

0

**0**

**Refractive index** *n(x)*

*nL*

lateral resonance condition for the first zone is given by

22 2

**First zone**

*ns1*

where

$$k\_o^2 = o^2 \,\varepsilon\_o \,\mu\_o \tag{16}$$

*ko*, *μo*, *ω* and *β* are the wave number, the permeability constant, the angular frequency and the allowed propagation coefficient along the *z*-axis, respectively.

Applying the continuity boundary conditions that connect the solutions at the interfaces *x = R* and *x = –R* and for *Δn<<no*, the solutions of the Eqs. (13)-(15) in the three regions (symmetric waveguide) are given in terms of Hermite polynomials (*Hq*) of degree *q*, as

$$E\_{y}(\mathbf{x}) = \begin{cases} A e^{-\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}R}{2}} H\_{q}(R\sqrt{\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}}{2R}}) e^{-\gamma(x-R)}, & \text{, } x > R \\\\ A e^{-\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}x^{2}}{2R}} H\_{q}(\sqrt{\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}}{2R}}), & -R < x < R \\\\ A e^{-\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}R}{2}} H\_{q}(-R\sqrt{\frac{k\_{o}(\Delta\varepsilon)^{\frac{1}{2}}}{2R}}) e^{\gamma(x+R)}, & x < -R \end{cases} \tag{17}$$

where *q* is an integer that identifies the mode. *Hq* is the appropriate Hermite polynomial defined by

$$H\_q(\mathbf{x}) = (-1)^q \ e^{\mathbf{x}^2} \quad \frac{d^q}{d\mathbf{x}^q} \ e^{-\mathbf{x}^2} \tag{18}$$

The first three Hermite polynomials in *x* are

$$\begin{aligned} H\_0(\mathbf{x}) &= \mathbf{1}, \\ H\_1(\mathbf{x}) &= \mathbf{2} \,\mathbf{x}, \\ H\_2(\mathbf{x}) &= \mathbf{4} \,\mathbf{x}^2 - \mathbf{2} \end{aligned} \tag{19}$$

Also *γ* in Eq. (17) is the attenuation coefficient in the clad region and 2 *n n <sup>o</sup>* . The allowed propagation coefficient and the effective index for every mode are given respectively by

$$\beta = \sqrt{k\_o^2 \varepsilon\_o - \frac{k\_o \sqrt{\Delta \varepsilon}}{R} \left(1 + 2q\right)}\tag{20}$$

and

$$m\_{\rm eff} = \frac{\beta}{k\_o} \,. \tag{21}$$

where *εo* is the dielectric constant at the center of the symmetric optical waveguide, and it is related to the refractive index by <sup>2</sup> *o o n* .

### **4.2 Asymmetric optical waveguide**

76 Advanced Holography – Metrology and Imaging

*ko*, *μo*, *ω* and *β* are the wave number, the permeability constant, the angular frequency and

Applying the continuity boundary conditions that connect the solutions at the interfaces *x = R* and *x = –R* and for *Δn<<no*, the solutions of the Eqs. (13)-(15) in the three regions (symmetric waveguide) are given in terms of Hermite polynomials (*Hq*) of degree *q*, as

( ) <sup>2</sup> ( ) <sup>2</sup>

*<sup>k</sup> Ae H R e x R R*

( ) <sup>2</sup> ( ) <sup>2</sup>

where *q* is an integer that identifies the mode. *Hq* is the appropriate Hermite polynomial

*R*

*<sup>k</sup> Ae H R e x R R*

( ) <sup>2</sup>

*<sup>k</sup> E x Ae H x RxR*

*q*

( ) ( 1)

*q q <sup>d</sup> Hx e e*

> 0 1

*H x Hx x Hx x*

( ) 1, () 2, () 4 2

 

2

Also *γ* in Eq. (17) is the attenuation coefficient in the clad region and 2 *n n <sup>o</sup>*

 

allowed propagation coefficient and the effective index for every mode are given

*n*

where *εo* is the dielectric constant at the center of the symmetric optical waveguide, and it is

*<sup>k</sup> k q <sup>R</sup>* 

*o*

*k* 

*q*

1

( ) ( ), <sup>2</sup>

*o x R*

1

1

*o x R*

(18)

(20)

. (21)

(19)

. The

( ) ( ), <sup>2</sup>

*q qx x*

*dx*

2

(16)

(17)

2 2 *o oo k* 

the allowed propagation coefficient along the *z*-axis, respectively.

*o*

*o*

*o*

2 2

<sup>2</sup> 1 2 *<sup>o</sup> o o*

*eff*

related to the refractive index by <sup>2</sup>

*o o n* .

The first three Hermite polynomials in *x* are

1 2

*k R*

1 2 2

2

*k R*

*k x <sup>R</sup> <sup>o</sup> y q*

1 2

( ) ( ) ( ), <sup>2</sup>

where

defined by

and

respectively by

On the other hand, to find the allowed TE modes and the propagation coefficients of the asymmetric GRIN optical waveguide, we assume that, the GRIN optical waveguide, as shown in Fig. 3, consists of two graded index zones; both have a Gaussian refractive index profile. The first zone is in the range *0 < x < b1*, and the second one is in the range *b2 < x < b4*. There is a very thin layer of constant refractive index *ns1* confined between the two zones in the range *b1 < x < b2*. The plane at *x=0* represents the interface between the surface of the first zone of the optical waveguide and the immersion liquid (of refractive index *nL*) and the plane at *x = b4* represents the interface between the second zone of the waveguide and the substrate, which has refractive index *ns*.

### **4.2.1 TE modes in the first zone of waveguide**

The refractive index profile for each zone is fitted using a Gaussian function which could be used to determine the allowed TE modes for the optical waveguide sample. The first zone of the waveguide in the range *0 < x < b1* is represented by the Gaussian refractive index profile given by

$$m(\mathbf{x}) = n\_{s\_1} + \Delta n\_1 \ e^{-\left(\frac{\mathbf{x}}{b}\right)^2} \qquad \qquad \qquad 0 < \mathbf{x} < b\_1 \tag{22}$$

where *b* is the depth of the first zone (between *0* and *b1*), and *n1* is the difference between the refractive indices at *x = 0* and *x = b1*. The conditions for the wave propagation along the first zone of the waveguide are: (1) a zigzag of the beam path and (2) the total phase change must be a multiple of 2*π*. Using WKB approximation (Marcuse, 1973; Mathey, 1996), the lateral resonance condition for the first zone is given by

$$2\int\_{0}^{h\_1} \sqrt{k\_o^2 \ n^2(\mathbf{x}) - \beta^2} \, d\mathbf{x} \quad -2\Phi\_{1\not\!2} - 2\Phi\_{1\not\!3} = 2\,\text{m}\,\pi \,, \quad 1 \le m \le M\tag{23}$$

where *β = koneff* is the propagation constant of the *m*th mode; *m* is an integer. *M* is the total number of modes, *ko = 2π/λ* is the wave-vector, *λ* is the wavelength, and *neff* is the effective

Fig. 3. The waveguide confined between liquid and substrate media, arranged such that the refractive index *n(x)* is larger than that of the surrounding media (*nL* ) and (*ns*).

Digital Holographic Interferometric Characterization of Optical Waveguides 79

optical field. The position of MO2 is precisely adjusted. These two beams are recombined at

2D transition Stage

M1

BS2 MO1

CCD

MO2

*d*

Optical cell

The mirror M2 in the digital holographic set-up reflects the reference beam. This mirror can be tilted to control the position of the reference beam on the CCD camera. The reference beam reaches the CCD camera at a small incident angle with respect to the propagation direction of the object wave. Therefore, the off axis holographic configuration is used (Wahba & Kreis, 2009a, 2009b, 2009c; Yassien et al., 2010). In addition, the mirror M2 in the digital holographic setup is mounted on a piezoelectric transducer (PZT), which acts as the phase shifting tool. It is positioned in the reference arm and enables us to obtain four phase shifted holograms. The phase shifting steps start with 0o and the holograms are shifted mutually by *π*/2. These phase shifted holograms are recorded by an Allied Vision Marlin F145B2 CCD camera with pixel pitch 4.65 µm × 4.65 µm and pixel numbers 1392, and 1040 in the horizontal and vertical

Phase shifting digital holographic interferometry is used to investigate the optical parameters of GRIN optical waveguide samples. Fresnel off–axis holograms have been produced by using a Mach-Zehnder holographic arrangement; see Fig. 4. A piezoelectric transducer (PZT), acting as a phase-shifting tool, is applied in the reference arm and helps us to obtain four phase-shifted holograms. Fig. 5 represents the phase shifted digital holograms for a GRIN optical fibre sample. The GRIN optical fibre sample is immersed in a liquid of refractive index 1.46, which is greater than that of its cladding. The phase of the hologram (Fig. 5(a)) is assumed to be zero, whereas the phase shifts of the holograms (Fig.

the beam splitter *BS2*, which is identical to *BS1*.

Laser S

Fig. 4. Digital holographic interferometric set-up.

M2

PZT

L BS1

**6.1 Refractive index profile of GRIN optical waveguide** 

5(b) to 5(d)) are *π*/2, *π*, and 3*π*/2; respectively.

directions, respectively.

**6. Results and discussion** 

mode index for light that propagates along the optical waveguide. *1/2* and *1/3* are the phase changes at the film-liquid interface and at the first and second zones interface, respectively. Generally these last quantities are approximated to *1/2 π/2* and *1/3 π/4* (Mathey, 1996). The effective indices (*neff*) for the first zone are obtained numerically by substituting Eq. (22) into Eq. (23).

### **4.2.2 TE modes of second zone waveguide**

Referring to Fig. 3, we suppose that, without loss of generality, the second region of the waveguide in the range *b2< x < b4* is divided into two asymmetric parts which have an individually Gaussian refractive index profile. The first and the second parts within the ranges are *b2< x < b3* and *b3< x < b4*, respectively. The refractive index profile in these two parts of the second zone of the waveguide is given by

$$m(\mathbf{x}) = \begin{cases} m\_{s\_1} + \Delta m\_2 \ e^{-\left(\frac{\mathbf{x} - \mathbf{x}\_c}{b}\right)^2} & , b\_2 < \mathbf{x} < b\_3 \\\\ m\_{s\_2} + \Delta n\_3 \ e^{-\left(\frac{\mathbf{x} - \mathbf{x}\_c}{b}\right)^2} & , b\_3 < \mathbf{x} < b\_4 \end{cases} \tag{24}$$

where *b* is the irradiation depth for both, the first part between *b2* and *b3*, and the second part between *b3* and *b4. n2* is the difference between the refractive indices at the interface between the first and the second parts at *x = b3* and that at *x = b2*, whereas *n3* is the difference between the refractive indices at the interface between the first and the second parts at *x = b3* and that of the substrate at *x = b4*.

Similarly, by analogy with Eq. (23) we have for the first and second parts

$$2\int\_{b\_2}^{b\_3} \sqrt{k\_o^2 \ n^2(\mathbf{x}) - \beta^2} \, d\mathbf{x} \quad -2\Phi\_{1\not\equiv 1} = 2\,\mathrm{m}\,\pi \quad \mathbf{1} \le \mathbf{m} \le M \tag{25}$$

$$2\int\_{b\_3}^{b\_4} \sqrt{k\_o^2 \ m^2(\mathbf{x}) - \beta^2} \, d\mathbf{x} \quad -2\Phi\_{1\not\!3} = 2m\pi \ , \quad 1 \le m \le M \tag{26}$$

since *1/2 = 0* at *xc*.

The effective indices (*neff*) for the two parts of the second zone are obtained numerically by using Eq. (24) in Eqs. (25) and (26).

### **5. Digital holographic set-up**

The digital holographic setup is a Mach-Zehnder system, see Fig. 4. The optical waveguide sample is immersed in a liquid of refractive index *nl* near the cladding refractive index *nclad* of the GRIN optical fibre or the refractive index of the substrate of the waveguide sample. An Abbe refractometer with an accuracy of ±0.0001 is used to measure the refractive indices of the immersion liquids. The collimated laser beam crosses the sample and passes through the microscope objective MO1 with magnification 10x and N.A. 0.25. An identical microscope objective MO2 is installed in the reference arm to eliminate the curvature of the

phase changes at the film-liquid interface and at the first and second zones interface,

(Mathey, 1996). The effective indices (*neff*) for the first zone are obtained numerically by

Referring to Fig. 3, we suppose that, without loss of generality, the second region of the waveguide in the range *b2< x < b4* is divided into two asymmetric parts which have an individually Gaussian refractive index profile. The first and the second parts within the ranges are *b2< x < b3* and *b3< x < b4*, respectively. The refractive index profile in these two

2

2

, ( ) ,

*n ne b xb*

*c*

*n ne b xb*

*c*

*x x b*

 

> *x x b*

 

where *b* is the irradiation depth for both, the first part between *b2* and *b3*, and the second part between *b3* and *b4. n2* is the difference between the refractive indices at the interface between the first and the second parts at *x = b3* and that at *x = b2*, whereas *n3* is the difference between the refractive indices at the interface between the first and the second

2 2 3

3 3 4

,

 

> 

*1/2* and

*1/2 π/2* and

*1/3 π/4*

(24)

*m mM* (25)

*m mM* (26)

*1/3* are the

mode index for light that propagates along the optical waveguide.

1

*s*

2

Similarly, by analogy with Eq. (23) we have for the first and second parts

1 3 2 () 2 2 , 1

1 3 2 () 2 2 , 1

*k n x dx*

*k n x dx*

The effective indices (*neff*) for the two parts of the second zone are obtained numerically by

The digital holographic setup is a Mach-Zehnder system, see Fig. 4. The optical waveguide sample is immersed in a liquid of refractive index *nl* near the cladding refractive index *nclad* of the GRIN optical fibre or the refractive index of the substrate of the waveguide sample. An Abbe refractometer with an accuracy of ±0.0001 is used to measure the refractive indices of the immersion liquids. The collimated laser beam crosses the sample and passes through the microscope objective MO1 with magnification 10x and N.A. 0.25. An identical microscope objective MO2 is installed in the reference arm to eliminate the curvature of the

22 2

22 2

*s*

respectively. Generally these last quantities are approximated to

substituting Eq. (22) into Eq. (23).

since 

**4.2.2 TE modes of second zone waveguide** 

parts of the second zone of the waveguide is given by

*n x*

parts at *x = b3* and that of the substrate at *x = b4*.

3

*b*

2

4

*b*

3

*b*

*b*

using Eq. (24) in Eqs. (25) and (26).

**5. Digital holographic set-up** 

*1/2 = 0* at *xc*.

*o*

*o*

optical field. The position of MO2 is precisely adjusted. These two beams are recombined at the beam splitter *BS2*, which is identical to *BS1*.

Fig. 4. Digital holographic interferometric set-up.

The mirror M2 in the digital holographic set-up reflects the reference beam. This mirror can be tilted to control the position of the reference beam on the CCD camera. The reference beam reaches the CCD camera at a small incident angle with respect to the propagation direction of the object wave. Therefore, the off axis holographic configuration is used (Wahba & Kreis, 2009a, 2009b, 2009c; Yassien et al., 2010). In addition, the mirror M2 in the digital holographic setup is mounted on a piezoelectric transducer (PZT), which acts as the phase shifting tool. It is positioned in the reference arm and enables us to obtain four phase shifted holograms. The phase shifting steps start with 0o and the holograms are shifted mutually by *π*/2. These phase shifted holograms are recorded by an Allied Vision Marlin F145B2 CCD camera with pixel pitch 4.65 µm × 4.65 µm and pixel numbers 1392, and 1040 in the horizontal and vertical directions, respectively.
