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
The effective index of an optical waveguide is an important parameter to determine the characteristics of guiding mode propagation. Knowing refractive index profiles of optical waveguides helps to design other complicated devices in integrated optical circuits. The interferometric techniques were popular in determining the effective index and index profile of the optical waveguides [1–3]. Nevertheless, destructive and laborious measurements were required when full index profiles were determined [1, 2]. Nondestructive methods, such as prism coupler, measure the effective indexes of a multimode waveguide, and the index profile can then be determined using the WKB approximation [4–7]. However, large number of modes was required for precise measurement. Another popular way to reconstruct the index profile is to measure distribution of the guiding mode . Iterative approach was adopted to solve the wave equation by changing the parameters of a known index model [9–11]. For the waveguide with unknown index profile, an inverse algorithm should be used [12–15]. Using a finite difference method based on the scalar wave equation, the refractive index profiles of optical waveguides can be reconstructed from the measured guiding mode intensity and its derivatives. Nevertheless, noises were amplified on performing numerical differentiation for the field-derivatives. Extra field smoothing technique was required in advance for the measured guiding mode profile. Numerical errors will occur due to the smoothing process. On the other hand, if the field distribution and its derivatives can be measured correctly enough, such problem can be overcome. Previously, we proposed a better and more accurate method to reconstruct the refractive index profiles of optical waveguides [8, 16]. The issue of noises amplified by the numerical differentiation was solved by measuring the differential fields directly. Nevertheless, this method cannot obtain the effective index and problems still occurred with the exact index value at the substrate region since the guiding fields at the substrate region were too weak, especially when two fields with low signal-to-noise ratios (SNRs) were divided by each other.
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, 18]. 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.
2.1 Inverse algorithm
For a weakly guiding waveguide with small index difference, the full index profile (n) can be expressed as the combination of substrate index (ns) and index difference (Δn), . Based on Helmholtz equation, , and the weakly guiding assumption, the refractive index profile is determined from an inverse algorithm [14, 16],8, 16]. The inverse algorithm for the refractive index reconstruction with measured optical differential fields can be expressed as Eq. (2).8]. However, large noises existed at the substrate region, due to the division of intensity and its derivatives (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 Δneff cannot be calculated from the 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) shows an example of a three-layer planar waveguide with index value ns, n2, and n1, where , and ns represents the substrate index. Consider the field distribution at the substrate region (), the theoretical evanescent distribution of the guiding mode I = E2 as well as its derivatives, Iy and Iyy, are described asEq. (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.
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 .
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 , 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 . 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.
In this work, a single mode fiber and two single-mode planar waveguides with different index profiles were measured for the demonstration. A step-index fiber (Thorlabs, SM600) with a numerical aperture (NA) of 0.11 was measured at 632.8nm. This fiber has a ~3.6μm core, 125μm cladding and 245μm coating. A proton-exchanged (PE) LiNbO3 waveguide was demonstrated for a three-layer waveguide due to its step-like index profile. It was fabricated by using a clean z-cut LiNbO3 substrate immersed in the benzoid acid at 210°C in a furnace and exchanged for 1hr. A titanium indiffusion lithium niobate (Ti:LiNbO3) 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 LiNbO3 substrate. To avoid the unwanted Li2O out-diffusion guiding layer, the sample was buried in lithium oxide powder in a crucible during the diffusion process in a furnace [21, 22], with diffusion temperature and time 1020°C and 3hrs, respectively. Natural cooling was done before the waveguide samples were taken out from the furnace. Both end-facets of previous mentioned waveguides were polished before end-fire coupling measurement.
A modified end-fire coupling method  was used for measuring the differential fields of the optical waveguides, as shown in Fig. 2 . 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.23]. 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 . 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) , 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) , 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) , 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) , 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) 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 . 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.
Using Eq. (2), the reconstructed index profile of the PE (green line) and Ti:LiNbO3 (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:LiNbO3 waveguide (blue line), the estimated maximum index difference is around 0.007, which is consistent for a Ti:LiNbO3 single-mode waveguide at the operating wavelength of 632.8nm . Index changes of these two waveguides are in different order, due to different fabrication process. As can be seen from Fig. 9, the diffusion depth of Ti:LiNbO3 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:LiNbO3). 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, 15, 25, 26]. This maybe overcome by adding a cladding layer, with index close or equal to the substrate, to top surface of the optical waveguide, using the wafer bonding technique [27, 28].
In conclusion, the effective indexes of a single mode fiber, a PE and a Ti:LiNbO3 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 , we have shown that the modified end-fire coupling method can reconstruct optical waveguides with slowly varying index profiles very well, especially in the core region. However, the lack information of neff 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 neff 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 neff to the calculated neff is 1.4582/1.4586. For the LiNbO3 waveguide, the fitted neff to the neff obtained by the prism coupling method is 2.2884/2.2890. The error is about 4~6 x10−4. 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.
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.
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