## Refractive index profiling of an optical waveguide from the determination of the effective index with measured differential fields |

Optics Express, Vol. 20, Issue 24, pp. 26766-26777 (2012)

http://dx.doi.org/10.1364/OE.20.026766

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### Abstract

The evanescent tails of a guiding mode as well as its first and second derivatives were measured by a modified end-fire coupling method. The effective index of the waveguide can be obtained by simultaneously fitting these three fields using single parameter. Combined with an inverse calculation algorithm, the fields with fitted evanescent tails showed great improvement in the refractive index profiling of the optical waveguide, especially at the substrate region. Single-mode optical fibers and planar waveguides of proton-exchanged (PE) and titanium-indiffusion (Ti:LiNbO_{3}) on lithium niobate substrates with different refractive index profiles were measured for the demonstration.

© 2012 OSA

## 1. Introduction

1. W. E. Martin, “Refractive index profile measurements of diffused optical waveguides,” Appl. Opt. **13**(9), 2112–2116 (1974). [CrossRef] [PubMed]

3. Y. Dattner and O. Yadid-Pecht, “Analysis of the effective refractive index of silicon waveguides through the constructive and destructive interference in a Mach-Zehnder interferometer,” IEEE Photonics J. **3**(6), 1123–1132 (2011). [CrossRef]

1. W. E. Martin, “Refractive index profile measurements of diffused optical waveguides,” Appl. Opt. **13**(9), 2112–2116 (1974). [CrossRef] [PubMed]

2. R. Oven, “Extraction of phase derivative data from interferometer images using a continuous wavelet transform to determine two-dimensional refractive index profiles,” Appl. Opt. **49**(22), 4228–4236 (2010). [CrossRef] [PubMed]

4. J. M. White and P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. **15**(1), 151–155 (1976). [CrossRef] [PubMed]

7. L. Wang and B.-X. Xiang, “Planar waveguides in magnesium doped stoichiometric LiNbO_{3} crystals formed by MeV oxygen ion implantations,” Nucl. Instrum. Meth. Phys. Res. Sect. B **272**, 121–124 (2012). [CrossRef]

8. W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. **36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

9. X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O^{2+} ion implantation into lithium niobate,” Opt. Commun. **281**(6), 1529–1533 (2008). [CrossRef]

11. G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive-index profiling of optical waveguides,” Appl. Opt. **35**(12), 2060–2068 (1996). [CrossRef] [PubMed]

12. D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett. **8**(2), 254–256 (1996). [CrossRef]

15. J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. **8**(5), 625–633 (1990). [CrossRef]

8. W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. **36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

16. W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. **91**(6), 061123 (2007). [CrossRef]

17. D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. **75**(8), 1039–1041 (1999). [CrossRef]

18. A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO_{3} waveguides,” Appl. Phys. Lett. **80**(13), 2239–2241 (2002). [CrossRef]

## 2. Algorithm

### 2.1 Inverse algorithm

*n*) can be expressed as the combination of substrate index (

*n*) and index difference (Δ

_{s}*n*),

14. I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol. **14**(3), 423–428 (1996). [CrossRef]

16. W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. **91**(6), 061123 (2007). [CrossRef]

*n*=

_{eff}*n*–

_{eff}*n*is defines as the effective index difference and

_{s}*k*

_{0}is the free space wavenumber. However, noises were amplified during the numerical differentiation (

8. W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. **36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

16. W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. **91**(6), 061123 (2007). [CrossRef]

*I*' and

*I*” represent the measured first and second order derivatives of

*I*, respectively. Previous work demonstrated refractive index profile reconstruction of optical fibers, especially at the guiding region [8

**36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

*I*” /

*I*and

*I*' /

*I*) with low SNRs, as can be seen from Eq. (2). The index profile was calculated only at the core region. Besides, only relative index difference was obtained. The absolute Δ

*n*cannot be calculated from the inverse algorithm.

_{eff}### 2.2 Evanescent wave fitting

*n*, the evanescent fitting method at the substrate region is applied. Figure 1(a) shows an example of a three-layer planar waveguide with index value

_{eff}*n*,

_{s}*n*

_{2}, and

*n*

_{1}, where

*n*represents the substrate index. Consider the field distribution at the substrate region (

_{s}*I*=

*E*

^{2}as well as its derivatives,

*I*and

_{y}*I*, are described aswhere

_{yy}*A*is the amplitude of the optical field and

*α*represents the decaying parameter of the exponential function,

*n*can then be obtained from the fitting parameter

_{eff}*α*, by fitting the evanescent waves to the exponential functions defined in Eq. (3). Simultaneous fitting of

*I*,

*I*and

_{y}*I*lowers the error compared with fitting with

_{yy}*I*only. More accurate results on

*α*and thus on

*n*can be expected. To check the

_{eff}*n*, we can replace the fitted evanescent waves of Eq. (3) into Eq. (2) for index reconstruction, the index difference (Δ

_{eff}*n*) at the substrate region should be zero, as expressed in Eq. (4), which matched with the physical nature of constant substrate index.The proposed multiple exponential fittings method can also overcome the noise problem of reconstructed index profile in the substrate region. Combining the inverse algorithm with the fitted evanescent tails, the large noises at the substrate region is eliminated.

*n*

_{1}and cladding index

*n*

_{2}, when the index contrast is very small, Δ = (

*n*

_{1}-

*n*

_{2})/

*n*

_{1}<<1, most power is confined within the core region for weakly guiding. The transverse mode can then be expressed analytically as Bessel functions [19].where

*k*

_{0}is the free space wavenumber and

*a*is the core radius.

*J*

_{0}(

*pr*) is the 0th-order Bessel function and

*K*

_{0}(

*qr*) is the modified Bessel functions of the second kind. The amplitude

*A*

_{0}is obtained by matching the continuity of tangential fields at the boundary

*r*=

*a*in cylindrical coordinate. By fitting the measured evanescent wave and its derivatives at the cladding region to

*K*

_{0}(

*qr*),

*K*

_{0}

^{'}(

*qr*) and

*K*

_{0}

^{”}(

*qr*), the effective index of the guiding mode,

*n*, can be obtained directly from the fitting parameters.

_{eff}*E*(

*y*) changes from negative to positive from waveguide core (

*n*(

*y*) =

*n*

_{2}>

*n*) to substrate (

_{eff}*n*(

*y*) =

*n*

_{s}<

*n*). Hence, if the second derivative can be measured, the interface position can then be determined. An example regarding to the field distribution and boundary is shown in Fig. 1(b). Assume a three-layer planar waveguide with waveguide thickness 2 μm and a constant index difference (Δ

_{eff}*n*) between waveguide and substrate of 0.005. The field distribution

*E*(

*y*) can be obtained from the slab waveguide equations [20]. Using

*I*(

*y*) =

*E*

^{2}(

*y*), the guiding mode intensity and its first-order and second-order derivatives (

*I*) were calculated numerically. As can be seen from Fig. 1(b), two peak values of

_{y}, I_{yy}*I*located at the air-waveguide and waveguide-substrate interfaces, where the maximum index differences occurred. There were two different analytical solutions for

_{yy}*I*at the boundaries (

*y*= 0 and

*y*= -

*t*). The opposite sign of the slope of

_{g}*I*at boundaries demonstrated the discontinuous nature of two analytical solutions. The waveguide-substrate boundary thus can be determined directly from the position of the local maxima of

_{yy}*I*. If the waveguide-substrate boundary is determined, then the effective index value can be calculated by fitting the measured wave and its derivatives (

_{yy}*I', I”*) to

*I, I*at the substrate region. The full index profile can then be completely reconstructed with the accurate effective index value.

_{y}, I_{yy}## 3. Experiment

_{3}waveguide was demonstrated for a three-layer waveguide due to its step-like index profile. It was fabricated by using a clean z-cut LiNbO

_{3}substrate immersed in the benzoid acid at 210°C in a furnace and exchanged for 1hr. A titanium indiffusion lithium niobate (Ti:LiNbO

_{3}) waveguide with a semi-Gaussian index distribution was also measured for the comparison. It was made by sputtering a 33nm-thick titanium film on a clean z-cut LiNbO

_{3}substrate. To avoid the unwanted Li

_{2}O out-diffusion guiding layer, the sample was buried in lithium oxide powder in a crucible during the diffusion process in a furnace [21

21. P. K. Wei and W. S. Wang, “A TE-TM mode splitter on lithium niobate using Ti, Ni, and MgO diffusions,” IEEE Photon. Technol. Lett. **6**(2), 245–248 (1994). [CrossRef]

**36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

*y*directions. A digital charge-coupled-device (CCD) camera was used to record images of the guiding mode vibration in a time sequence. The exposure time of CCD camera was chosen at 20ms for the measurement of unsaturated signals. 600 images were recorded as 12 cycles at the 1Hz vibrating frequency. Since these images were sinusoidally modulated at

*y*(or

*x*) axis with a small amplitude Δ

*y*(or Δ

*x*), by using Taylor expansion in Eq. (6), the first- and second- order differential fields can be obtained at the harmonic frequencies,

*ω*and 2

*ω*, respectively, as spatially vibrated modes.where ω is the angular vibrating frequency [23

23. C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. **67**(26), 3835–3837 (1995). [CrossRef]

## 4. Results and discussion

*I*, whereas the signal at the first (1

*ω*) and second (2

*ω*) harmonic frequency corresponds to

*I*(or

_{y}*I*) and

_{x}*I*(or

_{yy}*I*), respectively, as spatially vibrated mode. By performing inverse Fourier transform to the signals at the harmonic frequencies and adding the phase information,

_{xx}*I*,

*I*(or

_{y}*I*) and

_{x}*I*(or

_{yy}*I*) can be obtained. Figures 4(a) , 4(b), and 4(c) demonstrated the measured guiding mode and its derivatives in

_{xx}*x*direction of the single mode fiber. Figures 4(d), 4(e), and 4(f) show the corresponding guiding modes with modified Bessel function fitted at the cladding region by using Eq. (5).

_{3}. PE waveguide is known to be similar to a step waveguide. The measured guiding mode intensity and its derivatives extracted from vibrating harmonic frequencies and added with phase information are shown in Figs. 6(a) , 6(b) and 6(c). Figures 6(d), 6(e) and 6(f) show the corresponding one-dimensional intensity profiles of

*I*,

*I*and

_{y}*I*(red lines), together with the evanescent waves fitted to the exponential functions (blue lines), as defined in Eq. (3). The waveguide-substrate interface for curve fitting was determined at the right local maximum of

_{yy}*I*, as can be seen more clearly from Fig. 6(f). The evanescent waves of

_{yy}*I*,

*I*and

_{y}*I*were fitted simultaneously with the same fitting parameters

_{yy}*α*and

*A*using Eq. (3). The effective index (

*n*) obtained from fitted parameter

_{eff}*α*was 2.2157 for the PE waveguide, where the substrate index (

*n*) was assumed to be 2.2029 for the TM-polarized guiding on z-cut sample, and the effective index difference (Δ

_{s}*n*) was then calculated as 0.0128. The film thickness of the PE waveguide was 1.26μm, determined from the measured positions of two local maxima of

_{eff}*I*.

_{yy}_{3}single mode planar waveguide was also measured.

*I*,

*I*and

_{y}*I*were obtained by retrieving signals from the vibrating sequence at harmonic frequencies and added with phase information, as shown in Figs. 7(a) , 7(b), and 7(c). The corresponding one-dimensional plots of the measured differential fields (red lines) with evanescent waves fitted with exponential functions of Eq. (2) (blue lines) were shown in Figs. 7(d), 7(e), and 7(f). The effective index (

_{yy}*n*) obtained from fitted parameter

_{eff}*α*was 2.2884, while the substrate index (

*n*) was assumed to be 2.2874. The effective index difference Δ

_{s}*n*was then calculated as 0.001. The film thickness was determined to be 2.96μm.

_{eff}_{3}and PE single mode planar waveguides was shown in Figs. 8(a) and 8(b), respectively. For the Ti:LiNbO3 waveguide,

*n*measured with prism coupler was 2.2890, which is very close to the value 2.2884 obtained from our method. Good accordance can be seen. For the PE on LiNbO

_{eff}_{3}waveguide, a deviation occurred. The effective index obtained by prism coupling was 2.2412, while using our method was 2.2157. This is due to the degrade of PE waveguides, which has been reported previously [24

24. A. Yi-Yan, “Index instabilities in protonexchanged LiNbO_{3} waveguides,” Appl. Phys. Lett. **42**(8), 633–635 (1983). [CrossRef]

*n*had a maximum 20% degradation in time scale for the proton-exchanged waveguides on z-cut LiNbO

_{eff}_{3}with pure benzoic acid. The modified end-fire coupling and prism coupling measurements were taken at different times, the

*n*degradation thus caused the deviation.

_{eff}_{3}(blue line) planar waveguides are shown together in Fig. 9 . The large noise in the substrate region was smoothed out as a constant substrate index. Step-like index profile was obtained for the measured PE waveguide (green line). The maximum index difference of PE waveguide was estimated to be 0.0237, which is reasonable with our fabrication condition. For the Ti:LiNbO

_{3}waveguide (blue line), the estimated maximum index difference is around 0.007, which is consistent for a Ti:LiNbO

_{3}single-mode waveguide at the operating wavelength of 632.8nm [10

10. F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol. **16**(7), 1348–1353 (1998). [CrossRef]

_{3}waveguide stretches deeper into substrate, compared with the PE waveguide with higher refractive index difference, in order to obtain the single mode condition. It is noted that different from the optical fiber, there is a deviation for the index profile at the air-waveguide interface for the planar waveguides (PE and Ti:LiNbO

_{3}). This is due to edge-diffraction of light and imperfect polishing during the fabrication process. Moreover, since the index difference between air and waveguide is relatively large (~1.2), abrupt change cannot be reconstructed with the inverse method. Similar cases were also shown for index profile reconstruction of metal indiffusion waveguides in previous works [10

10. F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol. **16**(7), 1348–1353 (1998). [CrossRef]

15. J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. **8**(5), 625–633 (1990). [CrossRef]

25. I. Fatadin, D. Ives, and M. Wicks, “Accurate magnified near-field measurement of optical waveguides using a calibrated CCD camera,” J. Lightwave Technol. **24**(12), 5067–5074 (2006). [CrossRef]

26. F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO_{3} optical waveguides,” J. Appl. Phys. **78**(9), 5345–5350 (1995). [CrossRef]

27. Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO_{3} single crystals for optical waveguides,” Appl. Phys. Lett. **66**(12), 1484–1485 (1995). [CrossRef]

28. G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. **31**(7), 1054–1058 (2009). [CrossRef]

## 5. Conclusion

_{3}single mode planar waveguides were obtained by simultaneously fitting the evanescent waves of the measured field and differential fields to modified Bessel or exponential functions in the substrate region. The interfaces of air-waveguide and waveguide-substrate were directly determined at the local maxima of the second-order differential field. In the previous article [8

**36**(11), 2008–2010 (2011). [CrossRef] [PubMed]

*n*and noisy fields in the cladding region made index profile less accurate in the cladding part. In this work, we developed an effective approach with a higher accuracy to simultaneously solve the problems of

_{eff}*n*and index profile in the cladding part. Based on the evanescent nature of the optical field in the cladding part, we simultaneously fit the measured evanescent tails with the known evanescent distributions. This three-fields fitting method effectively enhances the fitting accuracy. For the single-mode fiber, the ratio of fitted

_{eff}*n*to the calculated

_{eff}*n*is 1.4582/1.4586. For the LiNbO

_{eff}_{3}waveguide, the fitted

*n*to the

_{eff}*n*obtained by the prism coupling method is 2.2884/2.2890. The error is about 4~6 x10

_{eff}^{−4}. This accuracy is comparable to the commercial prism coupling machine with the index accuracy of 0.0005. With the effective index values obtained from the fitting parameters, the full refractive index profiles of above optical waveguides were reconstructed more accurately, especially in the substrate region.

## Acknowledgments

## References and links

1. | W. E. Martin, “Refractive index profile measurements of diffused optical waveguides,” Appl. Opt. |

2. | R. Oven, “Extraction of phase derivative data from interferometer images using a continuous wavelet transform to determine two-dimensional refractive index profiles,” Appl. Opt. |

3. | Y. Dattner and O. Yadid-Pecht, “Analysis of the effective refractive index of silicon waveguides through the constructive and destructive interference in a Mach-Zehnder interferometer,” IEEE Photonics J. |

4. | J. M. White and P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. |

5. | K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. |

6. | P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” J. Mod. Opt. |

7. | L. Wang and B.-X. Xiang, “Planar waveguides in magnesium doped stoichiometric LiNbO |

8. | W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. |

9. | X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O |

10. | F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol. |

11. | G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive-index profiling of optical waveguides,” Appl. Opt. |

12. | D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett. |

13. | S. Barai and A. Sharma, “Inverse algorithm with built-in spatial filter to obtain the 2-D refractive index profile of optical waveguides from the propagating mode near-field profile,” J. Lightwave Technol. |

14. | I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol. |

15. | J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. |

16. | W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. |

17. | D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. |

18. | A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO |

19. | C. Yeh and F. I. Shimabukuro, “Optical fibers,” in |

20. | R. G. Hunsperger, “Theory of optical waveguides,” in |

21. | P. K. Wei and W. S. Wang, “A TE-TM mode splitter on lithium niobate using Ti, Ni, and MgO diffusions,” IEEE Photon. Technol. Lett. |

22. | M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc. |

23. | C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. |

24. | A. Yi-Yan, “Index instabilities in protonexchanged LiNbO |

25. | I. Fatadin, D. Ives, and M. Wicks, “Accurate magnified near-field measurement of optical waveguides using a calibrated CCD camera,” J. Lightwave Technol. |

26. | F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO |

27. | Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO |

28. | G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.3190) Image processing : Inverse problems

(130.3730) Integrated optics : Lithium niobate

(230.7390) Optical devices : Waveguides, planar

(290.3030) Scattering : Index measurements

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 5, 2012

Revised Manuscript: October 26, 2012

Manuscript Accepted: October 29, 2012

Published: November 13, 2012

**Citation**

Wan-Shao Tsai, San-Yu Ting, and Pei-Kuen Wei, "Refractive index profiling of an optical waveguide from the determination of the effective index with measured differential fields," Opt. Express **20**, 26766-26777 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26766

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### References

- W. E. Martin, “Refractive index profile measurements of diffused optical waveguides,” Appl. Opt.13(9), 2112–2116 (1974). [CrossRef] [PubMed]
- R. Oven, “Extraction of phase derivative data from interferometer images using a continuous wavelet transform to determine two-dimensional refractive index profiles,” Appl. Opt.49(22), 4228–4236 (2010). [CrossRef] [PubMed]
- Y. Dattner and O. Yadid-Pecht, “Analysis of the effective refractive index of silicon waveguides through the constructive and destructive interference in a Mach-Zehnder interferometer,” IEEE Photonics J.3(6), 1123–1132 (2011). [CrossRef]
- J. M. White and P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt.15(1), 151–155 (1976). [CrossRef] [PubMed]
- K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol.3(2), 385–391 (1985). [CrossRef]
- P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” J. Mod. Opt.33, 127–143 (1986).
- L. Wang and B.-X. Xiang, “Planar waveguides in magnesium doped stoichiometric LiNbO3 crystals formed by MeV oxygen ion implantations,” Nucl. Instrum. Meth. Phys. Res. Sect. B272, 121–124 (2012). [CrossRef]
- W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett.36(11), 2008–2010 (2011). [CrossRef] [PubMed]
- X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun.281(6), 1529–1533 (2008). [CrossRef]
- F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol.16(7), 1348–1353 (1998). [CrossRef]
- G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive-index profiling of optical waveguides,” Appl. Opt.35(12), 2060–2068 (1996). [CrossRef] [PubMed]
- D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett.8(2), 254–256 (1996). [CrossRef]
- S. Barai and A. Sharma, “Inverse algorithm with built-in spatial filter to obtain the 2-D refractive index profile of optical waveguides from the propagating mode near-field profile,” J. Lightwave Technol.27(11), 1514–1521 (2009). [CrossRef]
- I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol.14(3), 423–428 (1996). [CrossRef]
- J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol.8(5), 625–633 (1990). [CrossRef]
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