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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 26766–26777
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Refractive index profiling of an optical waveguide from the determination of the effective index with measured differential fields

Wan-Shao Tsai, San-Yu Ting, and Pei-Kuen Wei  »View Author Affiliations


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:LiNbO3) on lithium niobate substrates with different refractive index profiles were measured for the demonstration.

© 2012 OSA

1. Introduction

It is known that the guiding wave outside the waveguide core decays exponentially, and the effective index can be calculated directly from the evanescent wave distribution. The effective index measurement by taking the evanescent field on the waveguide surface has been demonstrated using the near-field scanning optical microscopy (NSOM) [17

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

18. A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002). [CrossRef]

]. However, expensive and complicated system setup was required for the NSOM system and the whole refractive index profile cannot be directly obtained. In this work, we develop an approach to simultaneously determine the effective index and the whole refractive index profile by combining the inverse algorithm and the evanescent tails fitting method. A simple optical setup with modified end-fire coupling method was used for the measurement of guiding mode and its differential fields. The position of the evanescent field was found can be determined directly from the peak value of the second derivative of the guiding field. The effective index was calculated by fitting the measured evanescent field and its derivatives. Combining with the inverse algorithm and the measured differential fields with fitted evanescent tails, the full refractive index profile was then reconstructed more accurately, especially at the substrate region.

2. Algorithm

2.1 Inverse algorithm

2.2 Evanescent wave fitting

To solve the problems of large noises at the substrate region and the unknown of Δneff, the evanescent fitting method at the substrate region is applied. Figure 1(a)
Fig. 1 (a) The slab waveguide. (b) Theoretical calculation of optical fields with analytical solutions. The arrow shows the beginning of the evanescent wave determined from the peak value of Iyy. I, I' and I” indicate the measured fields for the fitting.
shows an example of a three-layer planar waveguide with index value ns, n2, and n1, where n2>n1ns, and ns represents the substrate index. Consider the field distribution at the substrate region (0y), the theoretical evanescent distribution of the guiding mode I = E2 as well as its derivatives, Iy and Iyy, are described as
{I(y)=A2exp(2αy)Iy=I(y)=2αA2exp(2αy)Iyy=I(y)=4α2A2exp(2αy),0y,
(3)
where A is the amplitude of the optical field and α represents the decaying parameter of the exponential function, α=k0neff2ns2. neff can then be obtained from the fitting parameter α, by fitting the evanescent waves to the exponential functions defined in Eq. (3). Simultaneous fitting of I, Iy and Iyy lowers the error compared with fitting with I only. More accurate results on α and thus on neff can be expected. To check the neff, we can replace the fitted evanescent waves of Eq. (3) into Eq. (2) for index reconstruction, the index difference (Δn) at the substrate region should be zero, as expressed in Eq. (4), which matched with the physical nature of constant substrate index.
Δn14nsk02[4α212(2α)2]+Δneff=α22nsk02+Δneff=k02(neff2ns2)2nsk02+Δneffk02(2nsΔneff)2nsk02+Δneff=0,
(4)
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.

This method can also be applied to index profiling of circular step-index optical fiber. For an optical fiber with core index n1 and cladding index n2, when the index contrast is very small, Δ = (n1- n2)/n1<<1, most power is confined within the core region for weakly guiding. The transverse mode can then be expressed analytically as Bessel functions [19

19. C. Yeh and F. I. Shimabukuro, “Optical fibers,” in The Essence of Dielectric Waveguides (Springer, 2008).

].
{Ey=A0(J0(pr)J0(pa))raEy=A0(K0(qr)K0(qa))ra,
(5)
where p=k0n12neff2,q=k0neff2n22, k0 is the free space wavenumber and a is the core radius. J0(pr) is the 0th-order Bessel function and K0(qr) is the modified Bessel functions of the second kind. The amplitude A0 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 K0(qr), K0'(qr) and K0(qr), the effective index of the guiding mode, neff, can be obtained directly from the fitting parameters.

It should be noted that the position for the beginning of the evanescent tail is a key parameter for the fitting. From scalar wave equation, since 2E(y)=k02(n2(y)neff2)E(y), the second-order derivative of E(y) changes from negative to positive from waveguide core (n(y) = n2>neff) to substrate (n(y) = ns<neff). 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 (Δn) between waveguide and substrate of 0.005. The field distribution E(y) can be obtained from the slab waveguide equations [20

20. R. G. Hunsperger, “Theory of optical waveguides,” in Integrated Optics: Theory and Technology (Springer, 2009).

]. Using I(y) = E2(y), the guiding mode intensity and its first-order and second-order derivatives (Iy, Iyy) were calculated numerically. As can be seen from Fig. 1(b), two peak values of Iyy located at the air-waveguide and waveguide-substrate interfaces, where the maximum index differences occurred. There were two different analytical solutions for I at the boundaries (y = 0 and y = -tg). The opposite sign of the slope of Iyy 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 Iyy. If the waveguide-substrate boundary is determined, then the effective index value can be calculated by fitting the measured wave and its derivatives (I', I”) to I, Iy, Iyy at the substrate region. The full index profile can then be completely reconstructed with the accurate effective index value.

3. Experiment

A modified end-fire coupling method [8

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]

] was used for measuring the differential fields of the optical waveguides, as shown in Fig. 2
Fig. 2 Experimental setup of the modified end-fire coupling method.
. A He-Ne laser at 632.8nm was used as the light source. Light was coupled to the end-facet of waveguides via a single-mode fiber and two objective lenses. For the measurement of high-resolution images, an 100X oil-immersion objective lens was chosen for the output coupling. Both the input objective lens and the waveguide were fixed on micro-positioning stages for precise optical alignment. The output objective lens was put on a closed-loop piezoelectric nano-positioning stage, where a sinusoidal voltage wave was sent to from a function generator for periodically perturbing the optical path in x and 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 Δy (or Δx), by using Taylor expansion in Eq. (6), the first- and second- order differential fields can be obtained at the harmonic frequencies, ω and 2ω, respectively, as spatially vibrated modes.
I[y(t)]I[y(0)]+ΔyIy[y(0)]sinωt+14Δy2Iyy[y(0)]cos2ωt,
(6)
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]

]. By selecting signals at harmonic frequencies, this method also avoids unwanted noises inherently, without the need of designing extra spatial filters to smooth the measured fields.

4. Results and discussion

To retrieve the guiding mode and its derivatives, the recorded sequence of the vibrating optical intensity was first transformed into frequency domain, as shown in Fig. 3
Fig. 3 Fourier spectrum of the recorded sequence.
. The signal at zero frequency corresponds to non-vibrated guiding mode intensity, I, whereas the signal at the first (1ω) and second (2ω) harmonic frequency corresponds to Iy (or Ix) and Iyy (or Ixx), respectively, as spatially vibrated mode. By performing inverse Fourier transform to the signals at the harmonic frequencies and adding the phase information, I, Iy (or Ix) and Iyy (or Ixx) can be obtained. Figures 4(a)
Fig. 4 Measured intensities of the single mode fiber extracted from vibrating harmonic frequencies: (a) Intensity, I. (b) First-order differential field, Ix. (c) Second-order differential field, Ixx. Measured intensities with evanescent tails fitted with Bessel functions: (d) I, (e) Ix, (f) Ixx.
, 4(b), and 4(c) demonstrated the measured guiding mode and its derivatives in 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).

The one-dimensional plots of the measured intensities with evanescent tails fitting with modified Bessel functions of the second kind in the cladding region were shown in Figs. 5(a)
Fig. 5 One-dimensional profiles of (a) I (b) Ix (c) Ixx. (d) Three-dimensional colored surface of the reconstructed index profile of the single mode fiber Δn. (e) One-dimensional cross-sectional plot of Δn(x).
, 5(b), and 5(c). Using Eq. (2), two-dimensional index profile of the single mode fiber can be reconstructed from measured differential fields with evanescent tails fitted with Bessel functions, as shown in Fig. 5(d). Compared with the directly measured one (red line) and the known index model (green dashed line), as shown in the one-dimensional plot of Fig. 5(e), the large noise of the index profile in the substrate region was smoothed out as a constant substrate index (blue line). The reconstructed index profile shows a step-like distribution, especially in the cladding region, with a maximum index difference of 0.0047. It is noted that a more graded-like than step-like index profile was observed in the core region. This is due to the limit of optical resolution. Consider optical diffraction limit and the vibration amplitude of piezoelectric stage, the total resolution of our system is about 0.7μm, which is ~1/5 of fiber core size. Therefore, the reconstructed index did not reveal a flat top in the core region. Nevertheless, step-like behavior of reconstructed index profile was obvious when performing the evanescent tail fitting in the cladding region, and the corresponding effective index was also obtained. Although more graded-like than step-like index profile was obtain in the core region, index difference can still be estimated and compared with the known value. The core size was determined to be 3.67μm, and the effective index obtained from fitting parameter is 1.4582. From the specification of the fiber, it has a cladding index of 1.4570 at 633nm guiding wavelength, the maximum index difference is 0.0042 and NA is 0.11. The calculated effective index value is 1.4586. Our measured data match quite well with the parameters obtained from the specification of the fiber. Compared with other technique for determining the index profile of an optical fiber, our approach is simple, accurate, and nondestructive.

For the measurement of planar waveguides, we compared PE and Ti:indiffusion waveguides in LiNbO3. 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)
Fig. 6 Measured intensities of PE on LiNbO3 planar waveguide extracted from vibrating harmonic frequencies. (a) Intensity, I. (b) First-order differential field, Iy. (c) Second-order differential field, Iyy. Measured (red lines) profiles and measured fields fitted with exponential functions (blue lines) in y direction: (d) I (e) Iy (f) Iyy.
, 6(b) and 6(c). Figures 6(d), 6(e) and 6(f) show the corresponding one-dimensional intensity profiles of I, Iy and Iyy (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 Iyy, as can be seen more clearly from Fig. 6(f). The evanescent waves of I, Iy and Iyy were fitted simultaneously with the same fitting parameters α and A using Eq. (3). The effective index (neff) obtained from fitted parameter α was 2.2157 for the PE waveguide, where the substrate index (ns) was assumed to be 2.2029 for the TM-polarized guiding on z-cut sample, and the effective index difference (Δneff) 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 Iyy.

For further investigation, a Ti:LiNbO3 single mode planar waveguide was also measured. I, Iy and Iyy were obtained by retrieving signals from the vibrating sequence at harmonic frequencies and added with phase information, as shown in Figs. 7(a)
Fig. 7 Measured intensities of Ti:LiNbO3 planar waveguide extracted from vibrating harmonic frequencies. (a) Intensity, I. (b) First-order differential field, Iy. (c) Second-order differential field, Iyy. Measured (red lines) profiles and measured fields fitted with exponential functions (blue lines) in y direction: (d) I (e) Iy (f) Iyy.
, 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 (neff) obtained from fitted parameter α was 2.2884, while the substrate index (ns) was assumed to be 2.2874. The effective index difference Δneff was then calculated as 0.001. The film thickness was determined to be 2.96μm.

The effective indexes obtained from our methods were compared with those measured directly by a prism coupler (Metricon). The measured effective index of Ti:LiNbO3 and PE single mode planar waveguides was shown in Figs. 8(a)
Fig. 8 Measured effective indexes with prism coupler of (a) Ti:LiNbO3 and (b) PE on LiNbO3 single mode planar waveguides.
and 8(b), respectively. For the Ti:LiNbO3 waveguide, neff 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 LiNbO3 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 LiNbO3 waveguides,” Appl. Phys. Lett. 42(8), 633–635 (1983). [CrossRef]

]. In the literature showed that neff had a maximum 20% degradation in time scale for the proton-exchanged waveguides on z-cut LiNbO3 with pure benzoic acid. The modified end-fire coupling and prism coupling measurements were taken at different times, the neff degradation thus caused the deviation.

5. Conclusion

Acknowledgments

This work was supported by National Science Council, Taipei, Taiwan, under Contract No. NSC 100-2221-E-260 −019 and NSC-100-2120-M-007-006.

References and links

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]

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]

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]

5.

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]

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. 33, 127–143 (1986).

7.

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. 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 O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008). [CrossRef]

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]

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]

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. 27(11), 1514–1521 (2009). [CrossRef]

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]

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]

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 LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002). [CrossRef]

19.

C. Yeh and F. I. Shimabukuro, “Optical fibers,” in The Essence of Dielectric Waveguides (Springer, 2008).

20.

R. G. Hunsperger, “Theory of optical waveguides,” in Integrated Optics: Theory and Technology (Springer, 2009).

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]

22.

M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc. 135, 85–91 (1988).

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]

24.

A. Yi-Yan, “Index instabilities in protonexchanged LiNbO3 waveguides,” Appl. Phys. Lett. 42(8), 633–635 (1983). [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:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995). [CrossRef]

27.

Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 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]

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

  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]
  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]
  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]
  5. 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]
  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.33, 127–143 (1986).
  7. 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]
  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 O2+ ion implantation into lithium niobate,” Opt. Commun.281(6), 1529–1533 (2008). [CrossRef]
  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]
  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]
  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.27(11), 1514–1521 (2009). [CrossRef]
  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]
  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]
  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 LiNbO3 waveguides,” Appl. Phys. Lett.80(13), 2239–2241 (2002). [CrossRef]
  19. C. Yeh and F. I. Shimabukuro, “Optical fibers,” in The Essence of Dielectric Waveguides (Springer, 2008).
  20. R. G. Hunsperger, “Theory of optical waveguides,” in Integrated Optics: Theory and Technology (Springer, 2009).
  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]
  22. M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc.135, 85–91 (1988).
  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]
  24. A. Yi-Yan, “Index instabilities in protonexchanged LiNbO3 waveguides,” Appl. Phys. Lett.42(8), 633–635 (1983). [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:LiNbO3 optical waveguides,” J. Appl. Phys.78(9), 5345–5350 (1995). [CrossRef]
  27. Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett.66(12), 1484–1485 (1995). [CrossRef]
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