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

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

Digital Holographic Interferometric Characterization of Optical Waveguides 81

Fig. 6. Reconstructed interference phase modulo 2*π* from the phase shifted digital holograms

(a) (b) (a) (b)

Fig. 7. Unwrapped interference phase distribution with normalized background of the GRIN

(a) (b)

Fig. 8 displays the calculated core refractive index profile of the GRIN optical fibre sample. These refractive indices are fitted using a parabolic function which could be used to

optical fibre, (a) tilted and (b) perpendicular to *x*-axis.

determine the TE modes for the GRIN optical fibre.

of Fig. 5.

Fig. 5. Phase shifted digital holograms of the GRIN optical fibre sample at wavelength λ = 632.8 nm, (a) 0, (b) *π*/2, (c) *π*, and (d) 3*π*/2.

According to the phase shift algorithm, these four phase shifted holograms are used to produce the complex field in the hologram plane. From this field, the interference phase distribution is reconstructed by the convolution algorithm, and the result is shown in Fig. 6. In this figure we have the wrapped interference phase. The phase has been unwrapped and then the linear increase of the background phase from left to right has been approximated by a linear regression and subtracted from the unwrapped interference phase. The result with normalized background is shown in Fig. 7(a). This image represents the interference phase map of the GRIN optical fibre sample, whose interference phase varies across the sample but remains nearly constant along the optical fibre sample. This is due to the fact that the refractive index of each layer of the optical fibre sample remains constant along the optical fibre sample. One notes that the GRIN optical fibre sample is tilted inside the optical phase map, see Fig. 7(a). The tilt angle of the sample is small but it increases the error in the calculated mean values of optical phase differences along the fibre. Also, it might cause a broadening of the calculated mean value of the fibre radius. A simple algorithm is used to avoid this tilting and its corresponding errors. The optical phase differences across the fibre are shifted in each row of the phase map matrix. The optical interference phase differences along each layer of the optical fibre are shifted to be in the same column of the phase map matrix. After the shifting of all rows, the optical interference phase differences along the fibre appear to be perpendicular to *x*-axis. Fig. 7(b) represents the perpendicular optical phase differences along the GRIN optical fibre sample.

The modified interference optical phase map (Fig. 7(b)) is used to calculate the mean values of the interference optical phase differences across the GRIN optical fibre sample. Then the mean values of interference phase differences across the GRIN optical fibre in combination with the multilayer model are used to estimate the refractive index profile of the GRIN optical fibre. The refractive index profile of the GRIN optical fibre cladding is determined using the multilayer model.

(d)

Fig. 5. Phase shifted digital holograms of the GRIN optical fibre sample at wavelength λ =

According to the phase shift algorithm, these four phase shifted holograms are used to produce the complex field in the hologram plane. From this field, the interference phase distribution is reconstructed by the convolution algorithm, and the result is shown in Fig. 6. In this figure we have the wrapped interference phase. The phase has been unwrapped and then the linear increase of the background phase from left to right has been approximated by a linear regression and subtracted from the unwrapped interference phase. The result with normalized background is shown in Fig. 7(a). This image represents the interference phase map of the GRIN optical fibre sample, whose interference phase varies across the sample but remains nearly constant along the optical fibre sample. This is due to the fact that the refractive index of each layer of the optical fibre sample remains constant along the optical fibre sample. One notes that the GRIN optical fibre sample is tilted inside the optical phase map, see Fig. 7(a). The tilt angle of the sample is small but it increases the error in the calculated mean values of optical phase differences along the fibre. Also, it might cause a broadening of the calculated mean value of the fibre radius. A simple algorithm is used to avoid this tilting and its corresponding errors. The optical phase differences across the fibre are shifted in each row of the phase map matrix. The optical interference phase differences along each layer of the optical fibre are shifted to be in the same column of the phase map matrix. After the shifting of all rows, the optical interference phase differences along the fibre appear to be perpendicular to *x*-axis. Fig. 7(b) represents the perpendicular optical

**50 m** 

The modified interference optical phase map (Fig. 7(b)) is used to calculate the mean values of the interference optical phase differences across the GRIN optical fibre sample. Then the mean values of interference phase differences across the GRIN optical fibre in combination with the multilayer model are used to estimate the refractive index profile of the GRIN optical fibre. The refractive index profile of the GRIN optical fibre cladding is determined

632.8 nm, (a) 0, (b) *π*/2, (c) *π*, and (d) 3*π*/2.

(a)

(b)

(c)

phase differences along the GRIN optical fibre sample.

using the multilayer model.

Fig. 6. Reconstructed interference phase modulo 2*π* from the phase shifted digital holograms of Fig. 5.

Fig. 7. Unwrapped interference phase distribution with normalized background of the GRIN optical fibre, (a) tilted and (b) perpendicular to *x*-axis.

Fig. 8 displays the calculated core refractive index profile of the GRIN optical fibre sample. These refractive indices are fitted using a parabolic function which could be used to determine the TE modes for the GRIN optical fibre.

Digital Holographic Interferometric Characterization of Optical Waveguides 83

of 22oC. This sample is immersed in a fluid of *nL=*1.4917 perfectly matching the refractive index of the substrate. The photo-induced refractive index variations in the waveguide area can be divided into two zones. The refractive index in the first zone is high at the top surface and decreases gradually until reaching a minimum value, so that the refractive index distribution in the first zone can be classified as a graded index profile. The refractive index distribution in the second zone looks like Gaussian shape. The original homogeneous material surrounds the modified area of the PMMA slab, where the refractive index of the PMMA slab is *ns*. The cross sectional area of the waveguide is divided into *Q* layers. The refractive index of each layer is constant. The refractive index of the ith layer is *ni*, where *i=1,2,….,Q.* In addition, the thickness of each layer is constant. The refractive index variations extend in the polymer slab up to depth *b*. The layer thickness is equal to *b/Q*.

Fig. 10. The measured refractive index profile of fabricated GRIN optical waveguide.

phase difference *∆φi* across each layer is given by

 

The irradiated PMMA slab is immersed in a fluid and then it is used in the holographic setup. The object beam crosses the irradiated PMMA slab and then is superposed on reference beam. The produced interference pattern contains a field distribution related to the refractive index variations inside the modified area. The optical interference phase difference due to each layer depends on the refractive index *ni*. The optical interference

> , 1, 2, 3,........ <sup>2</sup> *<sup>i</sup> n na i i si Q*

where *ai* is the width of the *i*th layer of the waveguide, and λ is the wavelength of the used light beam. The refractive index of PMMA slab (non-irradiated area) *ns* is equal to the refractive index of the immersion liquid *nL*. This equation is valid in this case because the refractive index difference between the maximum and minimum values is very small in comparison with the refractive index difference of GRIN optical fibre. The calculated

refractive index profile of the fabricted optical waveguide is shown in Fig.10.

(27)

Fig. 8. The calculated refractive index profile of GRIN optical fibre.

Fig. 9. (a) The reconstructed 3D refractive index profile, (b) the projection of the 3D refractive index profile of GRIN optical fibre.

According to the symmetry of GRIN optical fibre, where the GRIN optical fibre consists of a large number of coaxial layers, all points of each layer have the same refractive index. Whereas, the 3D refractive index can be presented in spherical polar coordinates as *n(r,θ,ρ)=n0+Δn(r/R)2* with *r=0: R*, *θ=0:π* and *ρ=0:2π*, then a simple algorithm based on the assumptions of the multilayer model is used to reconstruct the refractive index profile across the GRIN optical fibre. Hence, Fig. 9(a) shows the three dimensional view of the refractive index across the GRIN optical fibre. The projection of the three dimensional refractive index profile is presented in Fig. 9(b).

In addition the same procedures are applied to a fabricated GRIN optical waveguide. Hence the UV irradiation lithographic method is used to inscribe an optical waveguide in a slab of PMMA. The waveguide sample is prepared at Λ = 248 nm, with laser pulse energy density 32 mJ/cm2, R (repetition rate) = 5 Hz, N (number of pulses) = 1250 pulse, and temperature

Fig. 8. The calculated refractive index profile of GRIN optical fibre.

(a) (b)

Fig. 9. (a) The reconstructed 3D refractive index profile, (b) the projection of the 3D

According to the symmetry of GRIN optical fibre, where the GRIN optical fibre consists of a large number of coaxial layers, all points of each layer have the same refractive index. Whereas, the 3D refractive index can be presented in spherical polar coordinates as *n(r,θ,ρ)=n0+Δn(r/R)2* with *r=0: R*, *θ=0:π* and *ρ=0:2π*, then a simple algorithm based on the assumptions of the multilayer model is used to reconstruct the refractive index profile across the GRIN optical fibre. Hence, Fig. 9(a) shows the three dimensional view of the refractive index across the GRIN optical fibre. The projection of the three dimensional

100

(b)

<sup>r</sup>m r <sup>m</sup>

In addition the same procedures are applied to a fabricated GRIN optical waveguide. Hence the UV irradiation lithographic method is used to inscribe an optical waveguide in a slab of PMMA. The waveguide sample is prepared at Λ = 248 nm, with laser pulse energy density 32 mJ/cm2, R (repetition rate) = 5 Hz, N (number of pulses) = 1250 pulse, and temperature

refractive index profile of GRIN optical fibre.

r m r m

(a)

n refractive index

refractive index profile is presented in Fig. 9(b).

of 22oC. This sample is immersed in a fluid of *nL=*1.4917 perfectly matching the refractive index of the substrate. The photo-induced refractive index variations in the waveguide area can be divided into two zones. The refractive index in the first zone is high at the top surface and decreases gradually until reaching a minimum value, so that the refractive index distribution in the first zone can be classified as a graded index profile. The refractive index distribution in the second zone looks like Gaussian shape. The original homogeneous material surrounds the modified area of the PMMA slab, where the refractive index of the PMMA slab is *ns*. The cross sectional area of the waveguide is divided into *Q* layers. The refractive index of each layer is constant. The refractive index of the ith layer is *ni*, where *i=1,2,….,Q.* In addition, the thickness of each layer is constant. The refractive index variations extend in the polymer slab up to depth *b*. The layer thickness is equal to *b/Q*.

Fig. 10. The measured refractive index profile of fabricated GRIN optical waveguide.

The irradiated PMMA slab is immersed in a fluid and then it is used in the holographic setup. The object beam crosses the irradiated PMMA slab and then is superposed on reference beam. The produced interference pattern contains a field distribution related to the refractive index variations inside the modified area. The optical interference phase difference due to each layer depends on the refractive index *ni*. The optical interference phase difference *∆φi* across each layer is given by

$$\frac{\Delta\phi\_i}{2\pi}\mathcal{A} = \left(n\_i - n\_s\right)a\_i \quad , \text{ i } i = 1, 2, 3, \dots \dots Q \tag{27}$$

where *ai* is the width of the *i*th layer of the waveguide, and λ is the wavelength of the used light beam. The refractive index of PMMA slab (non-irradiated area) *ns* is equal to the refractive index of the immersion liquid *nL*. This equation is valid in this case because the refractive index difference between the maximum and minimum values is very small in comparison with the refractive index difference of GRIN optical fibre. The calculated refractive index profile of the fabricted optical waveguide is shown in Fig.10.

Digital Holographic Interferometric Characterization of Optical Waveguides 85

Fig. 11. Effective indices against the mode number *q* for GRIN optical fibre sample.

Fig. 12. Plot of the first mode *(q=0)* calculated for GRIN optical waveguide sample.

*s s* 1 11

*s s* 1 12

respectively by

Now we can calculate the effective indices *neff* for each TE mode for an asymmetric (fabricated) optical waveguide. The guiding condition for the first zone is represented by

> *o n nn k*

*o n nn k* 

Also the guiding condition for the second zone, for the first and second parts are given

, (31)

(32)

presence of a varying index of refraction (Landau & Lifshitz, 1960).

Fig. 12 represents the first mode for the sample. Fig. 13 represents the second and the sixth modes. The oscillations are featured by an amplitude increasing with *x* approaching *<sup>i</sup>*

where *i* is the interface order between the modes. Landau and Lifshitz proved that the increase in wave amplitude, when the reflection point is reached, is a typical property in the

*<sup>t</sup> x* ,

### **6.2 Mode field distribution of GRIN optical waveguides**

The parabolic fitting of the refractive index profile for the investigated symmetric GRIN optical fibre core is shown in Fig. 8. The calculated optical parameters *no*, *Δn* with their standard error (*Sd*) and ∆ε are presented in table 1. The accuracy of the measurements of these optical parameters using the digital holographic method is increased in comparison to the accuracy of the multiple-beam Fizeau fringes and the two-beam interference Pluta polarizing microscope (Hamza et al., 2001).

The corresponding parameters <sup>o</sup> and <sup>c</sup> for the centre of the core and the cladding of the optical waveguide respectively are determined. The solutions must be oscillatory in the core radius of the optical fibre in order to represent guided modes.

The modes are guided as long as the condition

$$
\sqrt{\varepsilon\_o} > \frac{\beta}{k\_o} > \sqrt{\varepsilon\_o - \Delta \varepsilon} \quad , \tag{28}
$$

is satisfied.

The values of *β* can be calculated directly by using Eq. (20) for every mode number *q*. The coefficient *A* in Eq. (17) is related to the power carried in the core of the optical waveguide. The power is calculated by integrating the *z* component of the time averaged Poynting vector over the cross-sectional area of the waveguide:

$$S\_z = -\frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}.\vec{z}\right) \tag{29}$$

The average power in TE modes is

$$P\_z = -\frac{1}{2} \int\_{-\alpha}^{\alpha} \mathbf{E}\_y \cdot \mathbf{H}\_x^\* \, d\mathbf{x} = \frac{\beta}{2\alpha \rho\_o} \int\_0^{\alpha} \left| \mathbf{E}\_y \right|^2 \, d\mathbf{x} \, \tag{30}$$

The parameter *P* does not indicate the total power but the power per unit length in the *y* direction. The numbers of TE modes is related to the number of nodes, since the GRIN optical waveguide supports thirty nine modes, starting from *q* = 0 to 38. Each propagated mode has an effective index given by Eq. (21).


Table 1. The calculated optical parameters of GRIN optical fibre

Fig. 11 represents the effective indices for all modes of the GRIN optical waveguide sample. The values of the effective indices are confined between the core centre and clad refractive indices according to Eq. (21).

The parabolic fitting of the refractive index profile for the investigated symmetric GRIN optical fibre core is shown in Fig. 8. The calculated optical parameters *no*, *Δn* with their standard error (*Sd*) and ∆ε are presented in table 1. The accuracy of the measurements of these optical parameters using the digital holographic method is increased in comparison to the accuracy of the multiple-beam Fizeau fringes and the two-beam interference Pluta

optical waveguide respectively are determined. The solutions must be oscillatory in the core

*o k* 

 

The values of *β* can be calculated directly by using Eq. (20) for every mode number *q*. The coefficient *A* in Eq. (17) is related to the power carried in the core of the optical waveguide. The power is calculated by integrating the *z* component of the time averaged Poynting

2

2 2 *z yx <sup>y</sup>*

*P E H dx E dx*

The parameter *P* does not indicate the total power but the power per unit length in the *y* direction. The numbers of TE modes is related to the number of nodes, since the GRIN optical waveguide supports thirty nine modes, starting from *q* = 0 to 38. Each propagated

*no ∆n ∆ε*

*Sd* ±1.42441x10-4 ±3.04453x10-4

Table 1. The calculated optical parameters of GRIN optical fibre

1.48824 0.02974 0.088526

Fig. 11 represents the effective indices for all modes of the GRIN optical waveguide sample. The values of the effective indices are confined between the core centre and clad refractive

 

*o*

 

for the centre of the core and the cladding of the

*S E Hz <sup>z</sup>* (29)

. (30)

, (28)

**6.2 Mode field distribution of GRIN optical waveguides** 

radius of the optical fibre in order to represent guided modes.

and <sup>c</sup>

polarizing microscope (Hamza et al., 2001).

The modes are guided as long as the condition

*o o*

vector over the cross-sectional area of the waveguide:

<sup>1</sup> Re .

<sup>2</sup> <sup>1</sup> \*

The corresponding parameters <sup>o</sup>

The average power in TE modes is

indices according to Eq. (21).

mode has an effective index given by Eq. (21).

is satisfied.

Fig. 11. Effective indices against the mode number *q* for GRIN optical fibre sample.

Fig. 12 represents the first mode for the sample. Fig. 13 represents the second and the sixth modes. The oscillations are featured by an amplitude increasing with *x* approaching *<sup>i</sup> <sup>t</sup> x* , where *i* is the interface order between the modes. Landau and Lifshitz proved that the increase in wave amplitude, when the reflection point is reached, is a typical property in the presence of a varying index of refraction (Landau & Lifshitz, 1960).

Fig. 12. Plot of the first mode *(q=0)* calculated for GRIN optical waveguide sample.

Now we can calculate the effective indices *neff* for each TE mode for an asymmetric (fabricated) optical waveguide. The guiding condition for the first zone is represented by

$$n\_{s1} < \frac{\beta}{k\_o} < n\_{s1} + \Delta n\_1 \tag{31}$$

Also the guiding condition for the second zone, for the first and second parts are given respectively by

$$n\_{s1} < \frac{\beta}{k\_o} < n\_{s1} + \Delta n\_2 \tag{32}$$

Digital Holographic Interferometric Characterization of Optical Waveguides 87

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

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**8. Acknowledgment** 

**9. References** 

University for Higher studies & Research."

Vol. 24, pp. 4383-4386

Vol. 20, pp. 2787–2794

Vol. 3, pp. 385

Vol. 26, pp. 328

and

$$n\_s < \frac{\beta}{k\_o} < n\_s + \Delta n\_3 \tag{33}$$

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 and second parts are four and eight modes respectively.

Generally, the number of the modes and the values of the effective indices *neff* depend on two factors; the first one is the depth and the second one is *∆n*.

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