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• Abstract

SECTION 1

## Introduction

The design of any optical component requires detailed knowledge of the characteristics of the material used under differing optical conditions. One characteristic to consider is the change in refractive index of the material with wavelength, which corresponds to the dispersion as light propagates through the component. Within an optical waveguide system, this effect is caused by a combination of both the material itself and the geometry of the waveguide, leading to a total effect which can be observed by direct measurement of the waveguide. This information is critical when designing waveguides suitable for specific purposes, such as telecommunications components or nonlinear interactions.

The direct measurement of the refractive index of the waveguide at various wavelengths is considered here, utilizing a number of integrated Bragg grating structures. The method is efficient, with all data taken in a single measurement step. A Sellmeier fit is applied to the data, and the fit is used to calculate the measured dispersion of the waveguide. By empirically calculating the effect of waveguide dispersion in this specific case, we can infer both the waveguide and material dispersion components of the measured total dispersion. The zero dispersion wavelength (ZDW) of the waveguide may also be discussed accurately.

SECTION 2

## Background

The typical short length of planar waveguides means that their wavelength-dependent dispersion characteristics are difficult to measure using conventional techniques. While optical fiber dispersion is usually measured using either modulation phase shift methods or by time of flight [1], these techniques usually require a few kilometers of fiber and are measured over a spectral range of a few nanometers. They directly yield the group delay of the waveguide, introducing an additional calculation in order to obtain the refractive index profile. These approaches do not work well for planar waveguides due to their short (typically centimeter) waveguide lengths and, thus, yield imprecise results. Furthermore, for optical waveguides both material and waveguide dispersion contribute to the total dispersion, making it difficult to use conventional bulk glass index measurement techniques. In addition, the waveguide fabrication process (in our case ultraviolet writing) modifies the index properties of the base material.

The dispersive properties of shorter waveguides can be measured using existing techniques, mainly variations on the interferometric method introduced by Tateda et al. [2]. The simplest arrangement involves measuring the group delay of two arms of an equal path length interferometer, one arm containing the waveguide under test. Repeating the measurement at a variety of wavelengths allows the first-order chromatic dispersion to be obtained. Variations on this technique include the method presented by Okamoto et al. [3], where the polarization-dependent chromatic group delay of short LiNbO3 waveguides is observed. In addition, the white light interferometry methods [4] measure waveguide-dependent dispersion via the temporal (group delay) method or the spectral (phase or period of spectrally resolved interference fringes) technique. These techniques are accurate and precise, as long as longer waveguide lengths are used; however, the measured parameter is still the group delay, requiring an additional error compounding calculation in order to obtain the dispersive refractive index.

The method introduced in this paper involves measuring the few cms of waveguide available over a wide wavelength range in order to obtain information about the effective index of the waveguide. Several Bragg gratings are produced within the waveguide with central wavelengths over a large spectral range, the subsequent measurement of the reflected spectrum providing effective index data. The method is efficient, with all data taken in a single measurement step. The broadband nature of the technique and the direct effective index measurement implies the technique could potentially also be utilized in the design stages of parametric pair production in glass waveguides or fibers [5].

SECTION 3

## Experimental Details

The waveguide structure fabricated to demonstrate our new technique was produced in a photosensitive silica-on-silicon sample, fabricated using a flame hydrolysis deposition (FHD) technique. The wafer consists of a thermal oxide underclad (17 $\mu\hbox{m}$ thick), with FHD core and cladding layers deposited in turn. The 6.2-$\mu\hbox{m}$ core layer was made photosensitive by the addition of germanium and boron dopants, and a 15.6- $\mu\hbox{m}$ cladding layer was deposited in a secondary FHD step with boron and phosphorus dopants. The layers were refractive index matched by control of the germanium, boron, and phosphorus dopants in each layer. The core layer was positioned within the focus of the two converging UV beams in the direct grating writing (DGW) setup [6], and the sample was translated at constant speed in order to create waveguiding regions. Control of the periodic modulation of the beam and the sample translation speed allows Bragg gratings to be produced simultaneously within the waveguides.

The waveguide produced contained fifteen 1-mm-long weak-type $I$ Bragg gratings, each separated by 0.25 mm of index matched waveguide. Index matching of the waveguide and grating is critical so as to ensure the measured dispersion of the localized grating region is identical to that of a waveguide without grating. Index matching is achieved by controlling the speed of translation of the sample and the laser modulation, to ensure the same power density is present in the photosensitive core throughout fabrication. The Bragg gratings have peak reflectivities at wavelengths between 1275 nm and 1625 nm, spectrally spaced by approximately 25 nm. The small spot size of the DGW system allows this wide spectral range to be achieved by changing only the periodic modulation of the interference pattern [7]; no adjustments to the intersection angle were made, and all gratings were written sequentially within the waveguide in a single fabrication step. The gratings were apodized to yield a Gaussian spectral shape that is suitable for data fitting.

Characterization of the sample was performed using a combined multi-SLED source (Amonics ASLD-CWDM-5-B-FA), guided via fiber through a 3-dB fiber coupler, fiber in-line polarizer, and polarization maintaining pigtail. The characterization setup is shown in Fig. 1. The fiber in-line polarizer is designed such that the maximum polarizer transmission is aligned to the slow axis of the PM patchcord thereby ensuring single polarization across the broad bandwidth. All results presented here were measured in the TE polarization state; the same method could be used for the TM mode. The reflected signal was collected using an optical spectrum analyzer (OSA) (Ando AQ6317B), and is shown in Fig. 2. A Gaussian fitting algorithm was applied to the spectra, as well as the central wavelength of each grating obtained. These data were used alongside the Bragg equation for coupling into the counter-propagating mode TeX Source $$n = {\lambda_{B}m \over 2\Lambda}\eqno{\hbox{(1)}}$$ to calculate the effective index of the mode $(n)$, where $\lambda_{B}$ is the measured central wavelength of the Bragg grating, $m$ is a positive integer representing the order of the reflection from the grating, and $\Lambda$ is the grating period known precisely from the UV writing process.

Fig. 1. Schematic of optical setup used to characterize the waveguide. Light from a broadband source is polarized before it is launched into the sample. The reflected spectrum is collected using an optical spectrum analyzer. Black lines indicate single mode SMF-28 fiber. Blue lines indicate polarization maintaining fiber. Green line indicates electronic data connection.
Fig. 2. Plots of collected data. (Top) Launch power (red) and raw reflectivity data from waveguide (black). The noise observed near 1400 nm corresponds to OH absorptions. (Bottom) Reflectivity data normalized to background. The figure shows wide detuning range of the direct grating writing system with spectral range of over 300 nm.

The data collected using this measurement technique give a range of effective index values around the expected ZDW of the material. In order to obtain a more precise wavelength-dependent effective index profile, the second-order Bragg reflection was obtained for certain gratings. Using a modified characterization setup with a periodically poled lithium niobate (PPLN) frequency doubled 1550-nm femtosecond fiber laser, the second-order reflection spectra for the gratings at 1550 nm and 1575 nm were obtained. These spectra provided additional data points in the 780-nm to 790- nm region and allow significantly greater accuracy at shorter wavelengths.

SECTION 4

## Results

The resulting wavelength dependence of the effective index, including both the first- and second-order Bragg reflections, is shown in Fig. 3. The plot shows the fitted three-term Sellmeier equation to the measured data obtained using the least squares fit method TeX Source $$n^{2}(\lambda_{0}) - 1 = {b_{1}\lambda_{0}^{2} \over \lambda_{0}^{2} - a_{1}} + {b_{2}\lambda_{0}^{2} \over \lambda_{0}^{2} - a_{2}} + {b_{3}\lambda_{0}^{2} \over \lambda_{0}^{2} - a_{3}}\eqno{\hbox{(2)}}$$ where $n$ is the refractive index, $\lambda_{0}$ the wavelength, and $a_{i}$ and $b_{i}$ the Sellmeier coefficients corresponding to absorption oscillators in the UV and mid infrared [8]. Fig. 3 shows the difference in the Sellmeier curves obtained when the additional second-order reflections from the gratings are included in the calculation. Inclusion of the two second-order grating responses increases the total measured index by $1.4 \times 10^{-3}$, improving significantly the accuracy of the method in this region. In order to obtain an accurate Sellmeier relation, additional data points are necessary. In this instance, the number of data points obtained is limited by the sources available to us for characterization of the waveguides.

Fig. 3. Chromatic dispersion curve showing the measured effective index at each measured Bragg grating wavelength (crosses) and the corresponding fitted Sellmeier curves for the data sets, including the second-order reflections (blue) and excluding the second-order reflections (green). The Sellmeier coefficients for the inclusive (blue) curve are as follows: $a_{1} = 0.007498$, $a_{2} = 0.011035$, $a_{3} = 97.930$, $b_{1} = 0.71813$, $b_{2} = 0.401985$, and $b_{3} = 1.117181$. Note the variation between the fitted curves at the extremes.

To calculate the error associated with the Sellmeier fit, we calculate the standard deviation of the measured data points from the fitted Sellmeier curve. Assuming this error is equal across the entire wavelength band, the Sellmeier coefficients can be modified to take into account these errors. The three-term Sellmeier equation represents three absorption oscillators, at positions $a_{i}$ and with strength $b_{i}$. By refitting the Sellmeier curves to the maximum and minimum deviations from the original fitted curve and changing only the strength of the oscillators, we obtain Sellmeier curves representing the worst-case scenario of the fit. These data are shown in Fig. 4. The error is due to a number of factors related to the experimental and analytical stages of the process, including the accuracy of grating period fabrication and grating fitting errors. The standard deviation method described calculates the worst-case error in the Sellmeier fit and, thus, contains these experimental errors.

Fig. 4. Maximum and minimum standard deviations of the data from the fitted curve. Plot shows best fit (black), maximum standard deviation (blue), and minimum standard deviation (green).

In order to calculate the total measured dispersion of the waveguide, (3) was used [9]. Using the Sellmeier equation (2), and differentiating twice, we obtain the wavelength-dependent total dispersion of the waveguide $D_{m}$ TeX Source $$D_{m} = -{1 \over \lambda_{0}c}\left(\lambda_{0}^{2}{d^{2}n \over d \lambda_{0}^{2}}\right) \times 10^{9}\ \hbox{ps/km.nm}.\eqno{\hbox{(3)}}$$

The total dispersion of a single mode waveguide is comprised of two components: material dispersion and waveguide dispersion. The contributions of material and waveguide dispersion are not strictly additive, as discussed by Marcuse [10]. In our case, as we consider only the fundamental mode, the contribution of waveguide dispersion is expected to be small, and therefore, we assume that the sum of the material and waveguide contributions yield the total dispersion of the waveguide. We can calculate the waveguide dispersion contribution $D_{w}$ to the total dispersion of a specific waveguide using the method described by Ghatak and Thyagarajan [11] TeX Source \eqalignno{D_{w} \simeq&\ -{n_{2}\Delta \over 3\lambda_{0}} \times 10^{7}\left(V{d^{2}(bV) \over dV^{2}}\right)\ \hbox{ps/km.nm}&\hbox{(4)}\cr \Delta \equiv&\ {n_{1}^{2} - n_{2}^{2} \over 2n_{1}^{2}}.&\hbox{(5)}} where $n_{2}$ is the refractive index of the cladding, $n_{1}$ is the refractive index of the core, $\lambda_{0}$ is the wavelength, $V$ is the waveguide parameter, and $b$ is the normalized propagation constant of the waveguide. $\Delta$ is given by (5). The final component $V(d^{2}(bV)/dV^{2})$ can be calculated empirically by the following equation: TeX Source $$V{d^{2}(bV) \over dV^{2}} \simeq 0.080 + 0.549(2.834-V)^{2}.\eqno{\hbox{(6)}}$$

This relation gives us an approximation of the waveguide dispersion accurate to within 95% of the true value for a step index waveguide with $V$ parameter between 1.3 and 2.3. For our waveguide, these correspond to $\lambda$ between 944 nm and 1671 nm [11].

In Fig. 5, we plot the calculated waveguide dispersion contribution (black) alongside the measured total dispersion with experimental errors (blue, green, red). Also shown is the calculated material dispersion (cyan, magenta, yellow), assuming an additive relation between the waveguide and material dispersion components. The errors were carried through the calculation and are plotted alongside the data. The plot shows the small contribution of the waveguide dispersion to the total measured dispersion, as expected from the literature [10].

Fig. 5. Measured total dispersion and calculated waveguide and material dispersions with fit errors. Measured dispersion is shown in blue, with maximum and minimum errors in red and green, respectively. The empirically calculated waveguide dispersion component is shown in black. Using this information and the assumption of additive dispersion contributions, material dispersion (cyan) is shown, alongside calculated errors (magenta, yellow).

Additionally, information about the ZDW of the waveguide can be extracted from these data. The ZDW of the waveguide is 1220.5 nm ± 0.3 nm. The contribution due to waveguide dispersion causes this to shift by 17.5 nm from the equivalent material dispersion ZDW.

SECTION 5

## Conclusion

We have presented a method of utilizing integrated Bragg grating structures to measure the wavelength-dependent effective index of our silica-on-silicon waveguides. This information could be used in the calculation of higher order chromatic dispersion components for further applications in telecommunications or pulse broadening applications. The technique can potentially be applied to other photosensitive structures, such as chalcogenide glasses and flat fiber. The effective index data can be used in the design stages of fabrication of materials suitable for nonlinear optical interactions, such as spontaneous four wave mixing or parametric pair production in waveguides. By measuring second-order Bragg reflections, at essentially half the wavelength, we are able to improve the accuracy of the Sellmeier fit for near-visible wavelengths.

## Footnotes

This work was supported by the Engineering and Physical Sciences Research Council U.K. Corresponding author: H. L. Rogers (e-mail: hlr1g09@orc.soton.ac.uk).

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