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The simultaneous generation of second and third order dispersion is demonstrated using nonlinearly chirped silicon waveguide gratings. The nonlinearly chirped gratings are designed to generate varying signs and magnitudes of group velocity dispersion and dispersion slope. The design, fabrication, and experimental characterization of the silicon waveguide gratings are performed. Second order dispersion as high as −2.3 X 106 ps/nm/km and third order dispersion as high as 1.2 X 105 ps/nm2/km and as low as 1.2 X 104 ps/nm2/km is demonstrated at 1.55µm.
© 2013 Optical Society of America
1. Introduction
The propagation of short pulses relies on managing the group velocity dispersion in the propagation medium [1]. For most applications, the management of second order dispersion is of primary concern. Dispersion results in unwanted pulse broadening, but can also be harnessed for interesting nonlinear effects such as four-wave mixing [2,3], optical pulse compression [4,5], and supercontinuum generation [6]. The trend towards faster optical information processing entails the use of wavelength division multiplexing [7] and optical time division multiplexing [8]. For successful OTDM, optical pulses are expected to become narrower to accommodate higher data rates and hence more susceptible to higher order dispersion. The third order dispersion (TOD) length [9] is a measure of the propagation length beyond which third order dispersion in the propagation medium starts to become important. The TOD length is inversely proportional to the cube of the pulse width, and inversely proportional to the TOD of the medium. It follows that optical pulses with shorter temporal widths have a shorter TOD length. The issue of third order dispersion and its pulse broadening effects in single mode fibers first arose over three decades ago [10]. To resolve the issue of pulse broadening from dispersion, several compensators have been demonstrated for both second order [11,12] and third order dispersion [13–16] mostly using optical fiber platforms. Nanophotonics for integration of various information systems on a chip using the CMOS compatible Silicon on Insulator (SOI) platform provides the same advantages as CMOS in microelectronics – reduced cost, increased performance, compact components with complex functionalities. Because of the high index contrast of silicon with respect to its cladding and the fact that light is highly confined in the core, the group velocity dispersion of silicon waveguides can exceed three orders of magnitude compared to single mode optical fibers [17–21]. The proliferation of SOI based nanophotonics coupled with the need to support high data rates on this platform implies that both second and third order dispersion will become increasingly important. The TOD of silicon waveguides, has been characterized to be up to three orders of magnitude larger than that in single mode optical fibers [17,19]. In addition, SOI waveguides engineered to have close to zero second order dispersion would experience much more pronounced effects from TOD. This further strengthens the importance of dispersion engineering in photonic wires and forms a motivational basis of this research paper. In this manuscript, we demonstrate an integrated optical device capable of generating large third order dispersion. The device is implemented on a silicon-on-insulator platform, and designed to generate both second and third order dispersion simultaneously, so as to adequately compensate for both types of dispersion. Such dispersive elements have potential applications in TOD compensation, pulse shaping and nonlinear optical applications.
2. Design
The dispersion in a propagation medium can be represented by a Taylor series expansion of the propagation constant in a waveguide [9]:
where ω is the angular frequency. The last two terms of the equation represent the group velocity dispersion or equivalently, the second order dispersion, and third order dispersion respectively. The TOD can also be expressed as, . In addition, the group velocity dispersion can be re-expressed as a function of wavelength to give the GVD parameter, whereas the third order dispersion parameter, S, is found as the local dispersion slope of the dispersion vs. wavelength plot.Single mode fibers are characterized by second and third order dispersion, D = 17 ps/nm/km and S = 0.072 ps/nm2/km at a wavelength of 1.55µm [9]. Prior work in fiber Bragg gratings utilized gratings which were chirped in a square root function with respect to the z-location [16]. This was to ensure that the generated second order dispersion was normal and the dispersion slope was negative, so as to compensate for the inherent anomalous dispersion and positive dispersion slope in single mode fibers at 1.55µm. The dispersion and dispersion slope in silicon waveguides however, can vary significantly as their geometries change [18]. For example, silicon waveguides which are 430nm by 1.3µm have normal dispersion with a positive dispersion slope at a wavelength of 1.55µm. Therefore, a dispersive element which is characterized by anomalous dispersion and a negative dispersion slope is ideal for compensating for both the second and third order dispersion in such a waveguide. Conversely, a silicon waveguide which is 300nm by 300nm possesses anomalous dispersion and a negative dispersion slope at a wavelength of 1.5µm, and in this case, dispersive elements possessing normal dispersion and a positive dispersion slope would be ideal for compensating for both the second and third order dispersion.
We propose to design dispersive elements capable of compensating for both the second and third order dispersion of the aforementioned devices using a silicon-on-insulator device as shown in Fig. 1(a). The device consists of a silicon waveguide, 500nm in width and 250nm in height with sinusoidally modulated sidewalls [22–25] with a modulation amplitude of 50nm. It should be noted that alternate grating configurations may also be adopted to engineer similar types of dispersion [26,27], but the sinusoidal sidewall configuration is chosen for its ease of fabrication with single-step lithography. The grating is apodized along its length to eliminate group delay ripple as well as ripple within the pass band. The local period of the grating device at any location, z is given by Λ(z) = Λave + ∆Λ * z2., where Λave = 295 nm to ensure an operating wavelength close to 1550nm and ∆Λ describes the total variation in the grating period within the device. Therefore, the chirp at any point in the grating varies quadratically with z. It should be noted that z is a dimensionless parameter describing the distance between the start of the grating and the location within the grating. Qualitatively, each wavelength of light travels a differential distance into the waveguide grating. This effect arises because of the continuously changing period along the grating length. Each wavelength component will travel into the waveguide grating and be reflected at the point where the local grating period is Bragg matched with the wavelength of light [28].
Fig. 1 (a) Schematic of the device generating third order group velocity dispersion. Inset shows a scanning electron micrograph of a device. (b) 2D FDTD modeling for the grating reflectivity with inputs at port 2 and (c) Simulated group delay for different values of ∆Λ with the input at port 1 (dashed lines) and port 2 (solid lines). (d) Group delay (green – without apodization and red – with apodization) and reflecivity (blue – without apodization and black – without apodization) for ∆Λ = 4nm, launched from port 1. (e) Extracted values of D and S plotted as a function of ∆Λ for inputs at ports 1 and 2.
2D finite difference simulations were performed to study the ability of the device to generate dispersion and dispersion slope of a specific sign and magnitude on an incident E-field which is TE-polarized. Each device has a fixed length, L = 500µm, and varying values of ∆Λ = 2nm, 4nm, 7nm and 10nm. Apodization in the form of a Blackman function governed by the expression, is also applied to the grating sidewall modulation amplitude along the grating length to eliminate group delay ripple and ripple within the transmission band. It is observed from Fig. 1(b) that the reflectivity plots for the devices have no ripple, implying the effectiveness of the apodization function in eliminating ripple in both the reflectivity and group delay responses [Fig. 1(c)]. The group delay for each device is calculated using the spectral phase, φ(ω) obtained from FDTD simulations using the expression, . To further illustrate the effectiveness of the apodization function in reflectivity and group delay ripple suppression, Fig. 2(d) shows the reflectivity and group delay for an unapodized device with ∆Λ = 2nm and light launched from port 2. It is observed that significant ripple exists in both the reflectivity and group delay spectra. It is therefore evident that the apodization function is necessary for obtaining a good spectral response.
Fig. 2 Reflection spectrum for ∆Λ = 2nm, 4nm, 7nm and 10nm (port 2). Inset shows the transmission spectrum for ∆Λ = 2nm in linear units showing ~90% extinction.
In order to extract the values of D and S, the group delay generated by each device, τ(λ), is fitted with a quadratic function [14],
where τ0 is a constant group offset, and λ0 is the blue edge of the group delay. The extracted values of S and D (scaled with the grating length) as ∆Λ is varied are plotted in Fig. 1(d). It is observed from Figs. 1(c) and 1(d) that the generated TOD can be tailored in both sign and magnitude. Light launched into input 1 (see Fig. 1) would undergo anomalous dispersion and negative TOD, whereas light launched into input 2 will experience normal dispersion and positive TOD. A smaller magnitude of both D and S is generated for larger values of ∆Λ. Note that the form of chirp introduced in these devices generate D and S which are opposite in sign, and thus are ideal for simultaneous compensation in silicon waveguides possessing D and S which are also opposite in sign.3. Experimental characterization
In order to characterize the designed devices, several gratings with varying values of ∆Λ are fabricated. The devices are first patterned using electron-beam lithography. Reactive ion etching is used to define the grating structures followed by an overcoating of PMMA as the overcladding. PMMA is chosen as the overcladding as its refractive index of 1.49 is quite closely matched to that of silicon dioxide. A scanning electron micrograph of a typical device is shown in the inset of Fig. 1(a). Inverse tapers are used to terminate the waveguides at both ends of the gratings in order to enhance fiber – waveguide coupling efficiency. An amplified spontaneous emission source is first launched into an in-line fiber polarizer to select TE light. The light is then launched into the waveguide using a tapered fiber. For reflectionmeasurements, a circulator is used to reroute the reflected data for measurement. The spectral measurements are then performed using an optical spectrum analyzer.
Gratings with ∆Λ = 2nm – 10nm were fabricated and their reflection spectrum was characterized. Each device is composed of an input waveguide ~200µm in length, followed by a grating of length, 500µm, followed by an output waveguide ~200 µm in length. Figure 2 shows the reflection spectra of several grating devices with light being launched into port 1. An overall increase in the bandwidth of the gratings is observed as ∆Λ is increased. Further, it is observed that the longer wavelengths have a lower reflectivity overall. This effect is likely attributed to the fact that the grating period changes much more rapidly further along the z-axis by virtue of the quadratic relationship between the grating pitch and z. In gratings where the chirp is large, ie. the period of the gratings is distributed over a much larger range, the total bandwidth is increased, but at the expense of a lower absolute reflectivity [28]. Since the grating pitch corresponding to the longer Bragg wavelengths change much more rapidly as z increases, the effective coupling strength at longer wavelengths is reduced. A secondary contribution to the lower reflectivities at longer wavelengths is due to the inverse relation between coupling coefficient and wavelength [28].
The group delay characteristics of the different devices are extracted using the Fabry-Perot oscillations generated from the reflected data [16]. As shown in the inset of Fig. 2, the transmission of light through the gratings is ~10%, implying that close to 90% of incident light within the bandgap is reflected. Therefore, the Fabry-Perot oscillations will be dominated by mirrors which are formed from the air-waveguide boundary at the input waveguide and the point of reflection in the grating. The generated Fabry-Perot oscillations possess a free spectral range (FSR), ∆λ(λ) which varies as a function of wavelength according to the expression [29]
The differential lengths over which light of different wavelengths propagate provide the group delay, . The group index, ng used [see Fig. 3(a)] to extract ∆λ and ∆τ(λ) is calculated using the effective indices found from a fully vectorial beam propagation method. Sellmeier coefficients for silicon and silicon dioxide are used in order for both material and waveguide dispersion to be accounted for. The measured group delay for devices is shown in Figs. 3(b) and 3(c). Devices with ∆Λ = 2nm led to a group delay which were relatively flat with respect to wavelength and deviated quite significantly from the expected profile. This likely implies that the electron-beam process used to write the patterns were unable to resolve the continuous period changes in the small range from 295nm – 297nm along the length of the device. Devices with ∆Λ = 4nm, 7nm and 10nm had group delay profiles [Fig. 3(c)] which agreed well with the spectral characteristics in simulations.
Fig. 3 (a) Calculated group index as a function of wavelength used to extract the group delay. (b) Group delay and reflectivity for device with ∆Λ = 2nm with light launched from port 2. (c) Group delay vs. wavelength for light launched into Ports 1 and 2 and for different values of ∆Λ. Circles denote group delay for light launched from port 1 (black - ∆Λ = 4nm, blue, ∆Λ = 7nm, red, ∆Λ = 10nm) Diamonds denote group delay for light launched form port 2 (yellow - ∆Λ = 4nm, purple - ∆Λ = 7nm, green - ∆Λ = 10nm).
In addition, Fabry-Perot oscillations from a device generating normal dispersion and positive TOD is shown in Fig. 4(a). It is clear that the period of the oscillations decreases quite quickly with wavelength. As a further check of the origin of the Fabry-Perot oscillations in the measured spectra, the cavity length, L(λ) can be calculated. From Fig. 4(a), it is observed that the FSR obtained in our measurements range from 0.4nm – 1.05nm. Using Eq. (2), this range of values of L(λ) corresponding to this range of FSRs is between 200µm and 700µm. This range of values of L(λ) matches very well with the length between the start of the input waveguide and the start of the grating (~200µm) and that between the start of the input waveguide and the end of the grating (~700µm). It is important to note that the presence of secondary oscillations arising from the air-waveguide boundary at the output waveguide is largely precluded by the low transmission of light through the gratings (see inset of Fig. 2). Note also that all waveguide bends used in our devices have a radius of 50µm, and therefore have minimal effects on the measured dispersion [30].
Fig. 4 (a) Fabry Perot oscillations arising from a device with L = 500µm with light launched into Port 2. (b) Measured S and D for light launched into Ports 1 and 2.
The devices generate a second order dispersion and third order dispersion simultaneously. In order to extract the TOD coefficient from each of the devices, fits are applied according to Eq. (1). for each of the curves. The extracted values of D and S are plotted in Fig. 4(b). It is observed that both positive and negative values of D and S with varying magnitudes are generated. In addition, it is observed that the magnitude of D and S decreases as ∆Λ is increased in magnitude from −4nm to −10nm, in good agreement with trends observed in the modeling results. The measured values of S for ∆Λ = 10nm when light is launched from port 2 and port 1 are 1.2X104 ps/nm2/km and −1.4X104 ps/nm2/km respectively. The dispersion slope generated in each of these devices can be tailored to be either positive or negative depending on the dispersion characteristics of the waveguide in question. Silicon waveguides with dimensions of 430nm by 1.3µm have normal dispersion with a positive dispersion slope at a wavelength of 1.55µm. Therefore, light launched into port 1 of the device with ∆Λ = 2nm, 4nm or10nm will impart anomalous dispersion and a negative dispersion slope necessary for compensating for both the second and third order dispersion in such a waveguide. Conversely, a silicon waveguide which is 300nm by 300nm possesses anomalous dispersion and a negative dispersion slope at a wavelength of 1.5µm, and in this case, launching light into port 2 of the device with ∆Λ = 4nm, 7nm or 10nm would be ideal for compensating for both the second and third order dispersion. Note also that adjustment of the average pitch of the grating can be easily performed to accommodate operation at wavelengths other than 1.55µm.
4. Conclusions
The design, fabrication and experimental characterization of a dispersive element capable of generating second and third order dispersion simultaneously has been demonstrated. Tailoring the design parameters allows the generation of arbitrary signs and magnitudes of GVD and TOD – a feature which is ideal when use in the context of silicon-on-insulator waveguides or any other material platform where waveguide geometry can strongly influence its dispersion. Second order dispersion as high as −2.3 X 106 ps/nm/km and third order dispersion as high as 1.2 X 105 ps/nm2/km and as low as 1.2 X 104 ps/nm2/km is demonstrated at 1.55µm.
Acknowledgments
Support from the SUTD-MIT International Design center and SUTD-ZJU Collaborative Research Grant is gratefully acknowledged.
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