1. Introduction

Microring-resonator filters have attracted much attention for their promising applications as optical add-drop filters in wavelength-division-multiplexing (WDM) networks [1–2], and in optical interconnections. High-performance add-drop filters require a wide free-spectral-range (FSR), low loss, high in-band extinction, and a fast roll-off. To increase FSR while retaining a low loss, ring radii below 10µm are needed [3], along with a high index-contrast (>100%) for strong optical-field confinement. To achieve high in-band extinction and fast roll-off, high-order microring filters, with multiple series-coupled microrings, are required [4–5]. Figure 1(a) depicts a second-order microring filter configuration, where the coupling coefficients κ _0,1 can be engineered to realize various filter responses, such as maximally flat and Chebyshev filters [6]. However, the small device size, high index contrast, and multi-ring configuration present a number of fabrication challengers, as the optical response of such filters is extremely sensitive to nanoscale dimensional deviations [7]. One of the crucial issues in high-order microring filters is the resonant-frequency mismatch between rings, which markedly reduces the filter performance [8–12]. As shown in Fig. 1(b), in a second-order microring filter designed to achieve maximally flat response, the flatness and symmetry of the through-port response, the in-band extinction and the filter roll-off are degraded due to the mismatch of resonant frequencies of the two rings.

In this paper, we demonstrate that the frequency mismatch is caused primarily by the intrafield distortion in Scanning-Electron-Beam-Lithography (SEBL). The intrafield distortion of an SEBL system (Raith 150) was measured, and the resulting frequency mismatch simulated for second-order microring filters. The simulations show a good agreement with experimental result. More important, a simple method is proposed to correct the intrafield-distortion-induced frequency mismatch. By applying this correction, the average frequency mismatch was reduced from -8.6GHz to 0.28GHz in second-order microring filters.

 

Fig. 1. (a). Diagram of a second-order microring-resonator filter, κ _0,1 represent the coupling coefficients and f _0,1 represent the resonant frequencies. (b) Illustration of impact of frequency mismatch on the through-port response for a second-order filter (simulation). The simulation was done by conventional Transfer-Matrix-Method using the following parameters: κ ₀=0.0788, κ ₁=0.00187, f ₁=191THz (1569.5nm), f ₂=f ₁ +Δf.

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2. Intrafield distortion in SEBL and its impact on frequency mismatch

In SEBL, the electron beam is deflected by magnetic lenses within the writing field which is normally ~100µm square. The fields are then stitched together to form a larger area by precisely moving the laser-interferometer-controlled stage on which the substrate is mounted. To avoid severe field-stitching errors, microring resonators are usually written in a single field [3]. Intrafield distortion is due to systematic beam-deflection error within a single field, caused by electron-optics imperfection and digital-analog converter error. It exists in every SEBL system. Figure 2(a) illustrates the problem. The beam is supposed to expose point O(x ₀, y ₀) but the actual beam position is at point B(x _B, y _B). The displacement vector

is called intrafield distortion at position O(x ₀, y ₀), where Δx and Δy represent intrafield distortion in x and y direction, respectively. Intrafield distortion is position-dependent, that is, Δx and Δy are functions of the designed beam position (x ₀, y ₀). Intrafield distortion can be measured by use of a nanoscale metrology method [13]. Figure 2(b) shows the principle of intrafield-distortion measurement. A metal grid with a fine pitch, generated on a silicon substrate by interference lithography, was utilized as a metrological reference with long-range spatial coherence. The grid was scanned by the SEBL system in x and y directions, and the secondary-electron signals collected. The corresponding intrafield distortion, Δx and Δy, at various positions in the writing field was then determined by comparing the phase of the collected signal to that of an ideal grid signal [14], as illustrated by the inset of Fig. 2(b). Figure 3 shows the measured intrafield distortion in a 100µm×100µm writing field in a Raith 150 SEBL system. Each point in the field has an associated Δx and Δy. From Fig. 3(a), it is seen that the maximum distortion in x direction occurs at the right edge of the writing field. This is attributed to the “fly-back” of the beam on this edge. Similarly, the maximum distortion in the y direction appears at the lower edge of the writing field. The maximum total intrafield distortion is around 20nm, which occurs at the lower-left corner of the writing field.

 

Fig. 2. (a). Illustration of intrafield distortion in SEBL. (b) Illustration of how intrafield distortion is determined from the phase difference between the ideal grid signal (blue) and the actual measured signal (red).

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Based on the measurement of intrafield distortion, the actual beam position (x_B, y_B) can be approximated by a polynomial function of the designed beam position (x₀, y₀) as

where the coefficients a_n and b_n can be extracted by numerically fitting Eq. (2) to the distortion maps of Figs. 3(a) and 3(b). In this paper, the polynomial in Eq. (2) is approximated to the 4^th order.

 

Fig. 3. Intrafield distortion in (a) x and (b) y direction in a 100µm×100µm writing field of a Raith 150 SEBL system. The two dimensional quasi-periodic character of the distortion (i.e. the quasi-periodic peaks and valleys in the above distortion maps) is probably an indicator of digital-analog converter imperfection of the system.

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In the fabrication of microring filters using SEBL, the geometric parameters of the ring, such as ring radius and waveguide width, usually deviate from designed values due to intrafield distortion. The resonant frequency, which is very sensitive to dimensional variations, is accordingly changed. For instance, 1nm dimensional variation in radius and width will result in tens of GHz frequency shift, depending on the design of the microrings. In high-order microring filters, as the rings are located at different positions of the writing field, each ring experiences a different frequency shift because of the position-dependent nature of intrafield distortion. Therefore, frequency mismatch between the rings occurs. And the amount of this frequency mismatch varies with the position of the filter in the writing field.

3. Experimental demonstration

In this paper, second-order microring filters were used in experiment to show the impact of intrafield distortion on frequency mismatch and to demonstrate the correction of frequency mismatch. The cross section of the waveguides used in this experiment is shown in Fig. 4(a). A core layer of 400nm silicon-rich silicon nitride (SiN, refractive index n=2.18) on a 3µm SiO₂ cladding was used for the bus waveguides and rings, with air top and side cladding. The SiN layer was deposited by Low-Pressure-Chemical-Vapor-Deposition (LPCVD), providing a uniform thickness across the wafer. The widths of bus and ring waveguides were 700nm and 900nm, respectively. The ring radius was 8µm. The bus-ring and ring-ring gaps were 120nm and 380nm, respectively. Finite-Difference-Time-Domain (FDTD) simulation shows that, in this design, 1nm deviation in ring radius and width will cause -18.4GHz and -33GHz resonant-frequency shift of the filter, respectively. The fabrication process described in [9] was used, with a Raith 150 SEBL system defining the patterns. A scanning-electron-micrograph (SEM) of the fabricated second-order filter is shown in Fig. 4(b).

Second-order filters written at 10 different positions relative to the writing field were fabricated, and the frequency mismatch obtained from through- and drop-port spectra. For each position, six filters were fabricated in six different physical writing fields (but at the same position relative to each writing field) to get an average. Each of these filters took up a single writing field. The red dots in Figs. 5(a) and 5(b) show the measured average frequency mismatch when the filters are centered at (x, 0) and (0, y) in the writing field, respectively. The error bars indicate the frequency mismatch fluctuation caused by some random factors in fabrication. Based on Eq. (2), geometric deviations of the microrings and the corresponding frequency mismatch caused by intrafield distortion can also be simulated at various positions of the writing field. The simulation results are shown in Fig. 5 as continuous curves. It is seen that the simulation agrees well with the experiment, which demonstrates that intrafield distortion in SEBL is the major cause of frequency mismatch in high-order microring filters.

 

Fig. 4. (a). Diagram of waveguides cross-section. (b) Scanning-electron micrograph of the fabricated second-order microring filter, and details in bus-ring and ring-ring coupling regions.

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Fig. 5. Frequency mismatch between the two rings of second-order microring filters at various positions in the SEBL writing field: (a) the positions of the center of the second-order filter are at y=0 and various values of x, from x=-24 µm to x=+24 µm, and (b) the positions of the center of the second-order filter are at x=0 and various values of y from y=-24 µm to y=+24 µm. The inset diagrams illustrate the orientations of the filters in the two cases. The continuous lines are the simulation results. The red dots are measured frequency mismatch without intrafield-distortion correction, and the black squares are with correction.

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4. Correction of intrafield distortion

SEBL is the most widely used lithography tool in fabrication of micro-photonic devices because of its high resolution; however, as discussed above, intrafield distortion in SEBL degrades the performance of these devices. Here, we propose a simple method to correct the intrafield distortion in SEBL. As mentioned above, the designed beam position (x ₀, y ₀) and actual beam position (x _B, y _B) are related by function f(x, y) and g(x, y) described in Eq. (2). Thus, by pre-distorting the designed beam position to a modified position (x ₁, y ₁) in the layout, where (x ₁, y ₁) satisfies

then the actual beam position will return to the desired position (x ₀, y ₀). Hence, the intrafield distortion can be corrected.

Using this correction method, second-order filters at various positions in the writing field were fabricated. The black squares in Fig. 5 are the measured frequency mismatch after correction. The frequency mismatch is around 0, which validates the correction method. Figure 6 shows a typical result of the filter responses with and without intrafield-distortion correction. In this filter, frequency mismatch is reduced from -11.5GHz to -1.2GHz. Figures 7(a) and 7(b) illustrate the statistical distribution of the measured frequency mismatch of a number of filters, without and with intrafield-distortion correction, respectively. The multi-peak distribution in Fig. 7(a) indicates the position-dependent nature of the frequency mismatch. The distributions were fitted by a normal distribution, where µ represents the average frequency mismatch while the standard deviation σ corresponds to the fluctuation of frequency mismatch introduced by random fabrication errors. It can be seen that the average frequency mismatch is reduced from -8.6GHz to 0.28GHz by applying the correction method. Standard deviations σ in Figs. 7(a) and 7(b) are close, which is reasonable because σ depends on uncontrolled random variations in fabrication process. It should be possible to further reduce σ by optimizing the fabrication process.

 

Fig. 6. Through-port (solid) and drop-port (dash) responses of a second-order filter (centering at (12µm, 0) in the writing field) with (blue) and without (red) intrafield-distortion correction.

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Fig. 7. Statistics of frequency mismatch of a number of second-order filters at several positions in the SEBL writing field, where µ is the average frequency mismatch, and σ is the standard deviation of frequency mismatch: (a) without distortion correction and (b) with distortion correction. Note that the correction results in a reduction of µ from -8.6GHz to 0.28GHz.

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5. Conclusion

In this paper, we experimentally demonstrate that the frequency mismatch in high-order microring resonator filters is caused primarily by intrafield distortion in SEBL. The intrafield distortion map of an SEBL system was measured. The frequency mismatch of second-order microring filters centered at various positions in the SEBL writing field was simulated based on the distortion map, which is consistent with experimental results. A simple method is also proposed to correct the intrafield distortion. By use of this correction method, the average frequency mismatch in second-order microring filters was reduced from -8.6GHz to 0.28GHz.