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Abstract
We propose an on-chip mode converter via two cascaded Bragg reflection processes. A forward conversion between two guided modes can be achieved with the aid of an additional mode. The proposed structure is theoretically studied and simulated via the rigorous three-dimensional finite-difference time-domain (3D-FDTD) method. The bandwidth and central wavelength of the proposed mode converter can be adjusted according to our theoretical analysis and simulation results. By applying the similar design approaches as fiber Bragg gratings, conversion spectra with different shapes can be obtained. As an example, several mode converters with bandpass and sidelobe-reduced spectra are designed. We also investigate and verify the mode conversion by experiment. Therefore, the proposed method may pave a new path for the mode converters with desired conversion spectra.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic integrated circuits (PICs) have recently attracted great interests owing to its potential for next-generation optical communications and sensings [1]. On-chip manipulation of guided modes is an important technology for the PIC chips [2]. Various mode/polarization-division multiplexing (MDM/PDM) technologies have been developed to further enhance the on-chip transmission capacity with a single wavelength carrier, where mode conversions are required for the mode (de)multiplexers or mode filters [3]. Besides, mode conversions are also widely applied among the cavity biosensors [4], direction-selective structures [5,6], and all-optical switching [7].
Recently, the on-chip mode conversions have been widely studied. One of the methods achieves the mode conversion by designing waveguides with their shape gradually varied, such as waveguide tapers [8–10], asymmetric directional couplers [11,12], asymmetric Y-junctions [13–15], and Mach-Zehnder interferometers [16]. In such waveguides, wavefronts of the guided modes are gradually shaped when transmitted in the waveguide, and therefore, these structures usually have large footprint size. Another method is to utilize the structures with large-modulated refractive index, such as photonic crystals [17,18], metamaterials [19], plasmonics [20,21], and deeply-etched trenches [22,23]. Wavefronts of the guided modes are strongly modulated in these structures, resulting in very compact size. The third typical method is based on the waveguide grating structures where the effective refractive indices are periodically modulated, such as long period gratings [24,25] and Bragg gratings (BGs) [26–28]. In these structures, different modes are coupled with each other due to the modulation of refractive indices, and the phase match condition should be satisfied according to the coupling-mode theory. These devices are typically long with high wavelength-selectivity.
BGs have many advantages, and varieties of BGs-based photonic devices have been investigated such as narrowband filters, interleavers, add/drop multiplexers, and dispersion compensators [29–33]. The corresponding design theories and fabrication methods have been well developed with over one decade’s development. However, when BGs are used for mode conversion, only reflective mode conversion can be achieved, which limits its applications in mode conversion. In this paper, we propose an approach to achieve the forward mode conversion with bandpass spectrum via two cascaded Bragg reflection processes. We simulated the structure via the rigorous 3D-FDTD method [34]. The bandwidth and central wavelength of the conversion spectrum can be adjusted. Various conversion spectral shapes can be obtained by using the design approaches in fiber Bragg gratings. For example, mode converters with bandpass and sidelobe-reduced spectra are designed. The experiment was also carried out and the results agreed well with the theoretical results, and the fabrication error tolerance is also analyzed then.
2. Principle
Generally, for the backward coupling between the mode i and m, the phase matching condition must be satisfied and can be written as
where βi and βm are the propagation constants of the mode i and m, respectively. Kim is the grating vector. Here, βi = 2πni/λ, βm = 2πnm/λ, and Kim = 2π/Λim, where ni and nm are the weighted averaged effective refractive indices of mode i and m, respectively, in the grating region [35]. Λim is the period of the grating Gim. Hence, the mode coupling wavelength (Bragg wavelength) λim of the grating Gim is determined byThen, we consider the two cascaded gratings in a waveguide that supports three modes: i, j, and m. Two gratings Gim and Gmj are designed to have the same coupling wavelengths, i.e., λcouple = λim = λmj or (ni + nm)Λim = (nj + nm)Λmj. nj is the weighted averaged effective refractive index of mode j in the grating region. Λmj is the period of the grating Gmj.
When the mode i is launched at the left facet as shown in Fig. 1, it passes through the Gmj and gets to the Gim. Then, the mode i is backward coupled to the mode m by the Gim. The mode m is further backward coupled to the mode j by the Gmj and passes through the Gim. As a result, the forward conversion from the mode i to j can be achieved with the aid of the intermediate mode m. The overall phase match condition by combining both the conditions in the Gim and Gmj can be expressed as,
Fig. 1 Schematic principle of the proposed mode converter. Symbols i, j and m denote the mode i, j, and m, respectively.
Moreover, one condition must be satisfied for the proposed converter: reflection must be avoided when the mode i and j pass through the Gmj and Gim, respectively. Therefore, large-index-contrast waveguides such as silicon strip waveguides should be used to obtain a large index difference between the adjacent modes. Then, the coupling wavelength, λcouple = (ni + nm)Λim, is well separated from other coupling wavelengths such as the Bragg wavelength of the mode i in the grating Gim, i.e., λBragg = 2niΛim. Grating morphologies such as sidewall gratings [26], cladding-modulated gratings [36] and silicon-dioxide gratings [37] can be used for the proposed converter. In this paper, the surface grating is considered.
As an example, we demonstrate a mode converter in a silicon stripe waveguide which supports the first three order quasi-transverse electric (quasi-TE) modes. Figure 2(a) illustrates the proposed converter’s structure in silicon-on-isolator (SOI) for the conversion from the first higher-order quasi-TE mode (TE1) to the second higher-order quasi-TE mode (TE2). The silicon waveguide core layer is sandwiched by an upper silicon dioxide cladding and a buried oxide (BOX) layer as shown in Fig. 2(b). The converter is composed of two gratings of G02 and G01. Here the quasi-TE fundamental mode (TE0) works as an intermediate mode. When a TE1 with a specific wavelength is launched, it first passes through the G02 with little reflection until reaches the G01. Subsequently, the TE1 is reflected along with mode conversion to the TE0. The resultant TE0 is backward propagated to the G02. Then the mode conversion from the TE0 to TE2 occurs in the G02. Afterwards, the TE2 passes through the G01. Finally, the forward mode conversion from the TE1 to TE2 is realized. Figure 2(c) shows a schematic of the overall phase match condition.
Fig. 2 Schematic of (a) the proposed device structure for conversion from the TE1 to TE2, (b) a cross-section view of the waveguide of the G02, and (c) the overall phase matching condition.
3. Device design and simulation results
3.1 Design of Bragg gratings
Considering the surface grating of the silicon waveguide as a perturbation, the coupling mode theory can be used to design the grating structure. The coupling coefficient can be determined by the overlap integral of the electric field in the grating region between two modes. Generally, the coupling coefficient between mode i and j is given below [38],
Here, ω is the angular frequency, Ei and Ej are the normalized field distributions of mode i and j, respectively, and ε is the first order Fourier component of the grating. Therefore, to obtain high-efficient mode coupling, grating morphologies can be designed by comparing the electric field distribution of different modes. The design methods of G01, G02, and G12 are studied as follows with the waveguide width w = 1,200.0 nm, waveguide height h = 220.0 nm, grating depth d = 40.0 nm, and the thickness of both the cladding and buried SiO2 assumed infinite.Figure 3(a) shows a top view of the G01 for the coupling between the TE0 and TE1. As shown in Fig. 3(b), the electric amplitudes of the TE0 and TE1 are out of phase between left and right sections—section I and II. Therefore, the grating is also divided into two sections which are out of phase with each other. According to Eq. (4), the coupling coefficient is also calculated when the cross point between section I and II is moved from the left to right facet as shown in Fig. 3(c). The highest coupling coefficient is obtained when t/w is 0.50, where t is the width of section I.
Fig. 3 (a) Top view of the schematic of the G01 (red color corresponds to etched region). (b) The x-component of electric amplitudes of the TE0 and TE1. (c) Calculated coupling coefficients when the cross point between section I and II is moved from the left to right facet, i.e., t/w is varied from 0 to 1.0.
Figure 4(a) shows a top view of the G02 for the coupling between the TE0 and TE2. As shown in Fig. 4(b), the electric amplitudes of the TE0 and TE2 can be divided into three sections—section I, II, and III. The center section is out of phase to the two outer sections. As shown in Fig. 4(c), the coupling coefficient is also calculated when the cross point between section I and II is moved from the left facet to the middle of the waveguide. At the same time, the cross point between section II and III is moved simultaneously in an opposite direction to maintain the grating symmetry. The highest coupling coefficient is obtained when t/w is 0.32, where t is the width of section I.
Fig. 4 (a) Top view of the schematic of the G02 (red color corresponds to etched region). (b) The x-component of electric amplitudes of the TE0 and TE2. (c) Calculated coupling coefficients when the cross point between section I and II is moved from the left to middle of the waveguide, i.e., t/w is varied from 0 to 0.5.
Figure 5(a) shows a top view of the G12 for the coupling between the TE1 and TE2. As shown in Fig. 5(b), the electric amplitudes of the TE1 and TE2 can be divided into four sections—section I, II, III, and IV. As a result, gratings in any two neighboring sections are out of phase with each other. The coupling coefficient is also calculated when the cross point between section I and II is moved from the left facet to the middle of the waveguide, as shown in Fig. 5(c). At the same time, the cross point between section III and IV is moved simultaneously in an opposite direction to maintain the grating symmetry. The highest coupling coefficient is obtained when t/w is 0.36, where t is the width of section I.
Fig. 5 (a) Top view of the schematic of the G12 (red color corresponds to etched region). (b) The x-component of electric amplitudes of the TE1 and TE2. (c) Calculated coupling coefficients when the cross point between section I and II is moved from the left to middle of the waveguide, i.e., t/w is varied from 0 to 0.5.
The central wavelength λ0 can be adjusted according to Eq. (2). The full bandwidth of the mode conversion spectrum can be determined by changing the coupling coefficient κ and total grating length L. The full bandwidth between nearest nulls of a grating Gij can be described as [11]:
where Lij denotes the grating length and ng,i and ng,j are the group indices of mode i and j, respectively. By applying the multimode transfer matrix method (MTMM) [32], the grating spectra of G01 are studied with varied κ and L as shown in Fig. 6(a). The full bandwidths are 2.1 nm, 4.0 nm, and 7.2 nm when κ = 100.0 cm−1 and L = 700.0 μm, κ = 200.0 cm−1 and L = 350.0 μm, κ = 350.0 cm−1 and L = 200.0 μm, respectively. The full bandwidth of G01 is calculated with varied κ and L when the product κL is fixed at 1.0, 3.0, 7.0, and 10.0, respectively, as shown in Fig. 6(b). It shows that the full bandwidth is increased with κ and decreased with L when the product κL is fixed.Fig. 6 (a) The simulated reflection spectra of G01 when the κ and L are varied. (b) The full bandwidth of G01 with respect to the κ and L when the product κL is fixed at the value of 1.0, 3.0, 7.0, and 10.0.
The side lobes of the BGs can cause unwanted crosstalk, which can be found in Fig. 6(a). The apodization structure is usually applied to suppress the side lobes [39,40]. The apodization is realized by changing the κ. Figure 7(a) shows two approaches for changing the κ—to change the transverse filling factor (TFF) or the longitudinal filling factor (LFF). The TFF is defined as the ratio of the grating width after apodization to that before apodization, i.e., w1/w0. The LFF is defined as the ratio of the grating length to the grating period, i.e., l1/l0. As shown in Fig. 7(b), the coupling coefficient shows a sine-shape curve when the LFF is varied from 0 to 1.0. The coupling coefficient increases with TFF when the TFF is changed from 0 to 1.0. Figure 7(c) shows an apodization of grating when the LFF is changed from 0 to 1.0 along the position. Therefore, the overall grating is apodized with a sine-shape index modulation, which can reduce the side lobes effectively.
Fig. 7 (a) Top view of the schematic of the TFF and LFF in G01 (red color corresponds to etched region). (b) The coupling coefficient with respect to the TFF and LFF. (c) An apodization of grating when the LFF is changed from 0 to 1.0 along the position.
3.2 Simulation results of G01, G02, and G12
Based on the design methods studied in Section 3.1, three types of gratings—G01, G02, and G12—are simulated via the rigorous 3D-FDTD method. The simulation parameters of the stripe waveguide are the same as that in Section 3.1. All three types of gratings are applied with the optimized design in Figs. 3-5 and the apodizations in Fig. 7(c). In our simulation, the averaged effective indices in the grating region are considered to determine the grating period according to Eq. (2). All the conversion wavelengths of these three types of gratings are all designed at 1,550.0 nm.
The grating period of the G01 is 300.1 nm, and the total number of periods is 200, which corresponds to the whole length of 60.0 μm. With the TE0 incident, the simulated reflection spectra of the TE0, TE1, and TE2 are shown in Fig. 8(a). The 3-dB bandwidth of the reflection spectrum to TE1 is 29.5 nm. The side lobes are reduced to be lower than −17.0 dB. The modal crosstalks from the TE0 and TE2 are lower than −23.5 dB. In the proposed mode converter, another important requirement of the G01 is the transmission property of the TE2. As shown in Fig. 8(b), the transmission loss of the TE2 is lower than 0.2 dB at the wavelength near 1,550.0 nm and the transmission modal crosstalks from the TE0 and TE1 are lower than −20.0 dB. No other modal crosstalks are observed in the wavelength range from 1,500 to 1,600 nm.
Fig. 8 Modal analyses of (a) reflection spectra of G01 with the TE0 incident, (b) transmission spectra of G01 with the TE2 incident, (c) reflection spectra of G02 with incident TE0, (d) transmission spectra of G02 with the TE1 incident, (e) reflection spectra of G12 with the TE1 incident, and (d) transmission spectra of G12 with incident TE0.
The grating period of the G02 is 326.6 nm, and the total number of periods is 200, which corresponds to the whole length of 65.3 μm. With the TE0 incident, the simulated reflection spectra of the TE0, TE1, and TE2 are simulated and shown in Fig. 8(c). The 3-dB bandwidth of the reflection spectrum to the TE2 is 24.5 nm. The side lobes are reduced to be lower than −16.5 dB. The modal crosstalks from the TE0 and TE2 are lower than −21.0 dB. In the proposed mode converter, another requirement of the G02 is the transmission property of the TE1. As shown in Fig. 8(d), the transmission loss of the TE1 is lower than 0.3 dB at the wavelength near 1,550.0 nm. The transmission modal crosstalks from the TE0 and TE2 are lower than −50.0 dB. In the wavelength near 1,588.0 nm, the Bragg reflection of the TE1 is observed. Owing to a 35-nm wavelength spacing and good apodization of sidelobes, this reflection will introduce little modal crosstalk in G02. However, when the effective refractive index difference is reduced between the n0 + n2 and 2n1, this will cause a large modal crosstalks here.
As for the G12, the grating period is designed to be 343.8 nm, and the total number of periods is 200, which corresponds to the whole length of 68.8 μm. With the TE1 incident, the simulated reflection spectra of the TE0, TE1, and TE2 are shown in Fig. 8(e). The 3-dB bandwidth of the reflection spectrum to the TE2 is 23 nm. The side lobes are reduced to be lower than −16.0 dB. The modal crosstalks from the TE0 and TE2 are lower than −24.0 dB. The transmission modal crosstalk from the TE1 is about −18.0 dB and that to the TE2 is lower than −38.0 dB. As shown in Fig. 8(f), a small undesired notch about −0.4 dB of the TE0 near 1565.0 nm is observed. This notch is caused by the weak coupling between the TE0 and the fundamental magnetic mode (TM0), which has been specifically studied in [38]. It can cause a small modal crosstalk in the reflection of G12.
3.3 Simulation results of overall forward mode conversions
Based on the individual grating sections of G01, G02, and G12 as analyzed in Section 3.2, two gratings from them are chosen to achieve one of the three kinds forward mode conversions—from TE0 to TE1, TE1 to TE2, and TE2 to TE0. The grating parameters are the same as those in Section 3.2.
The mode conversion from the TE2 to TE0 can be achieved with a combination of the G01 and G12 (G01-G12). As shown in Fig. 9(a), the x-component intensity of electric field |Ex| is simulated at the wavelength of 1,550.0 nm, which shows a mode conversion from the TE2 to TE0. Figure 9(b) shows the simulated conversion spectrum with the TE2 incident. The side lobes are lower than −20.0 dB. The conversion spectrum from the TE2 to TE0 has a 3-dB bandwidth of 20.2 nm. Modal crosstalks from the TE1 are lower than −19.7 dB. The peak transmission of the mode conversion is −0.2 dB.
Fig. 9 (a) Simulated x-component intensity of electric field at the wavelength of 1,550.0 nm, and (b) modal analyses of the transmission spectra with the TE2 incident in the structure of G01-G12.
The mode conversion from the TE0 to TE1 can be achieved with a combination of the G12 and G02 (G12-G02). As shown in Fig. 10(a), the x-component intensity of electric field |Ex| is simulated at the wavelength of 1,550.0 nm, which shows a mode conversion from the TE0 to TE1. Figure 10(b) shows the simulated conversion spectrum with the TE0 incident. The side lobes are lower than −29.8 dB. The conversion spectrum from the TE0 to TE1 has a 3-dB bandwidth of 23.5 nm. Modal crosstalks from the TE2 are lower than −22.5 dB. The peak transmission of the mode conversion is −0.2 dB.
Fig. 10 (a) Simulated x-component intensity of electric field at the wavelength of 1,550.0 nm, and (b) modal analyses of the transmission spectra with incident TE0 in the G12-G02.
Mode conversion from the TE1 to TE2 can be achieved with a combination of the G02 and G01 (G02-G01). As shown in Fig. 11(a), the x-component intensity of electric field |Ex| is simulated at the wavelength of 1,550.0 nm, which shows a mode conversion from the TE1 to TE2. Figure 11(b) shows the simulated conversion spectrum with the TE1 incident. The side lobes are lower than −36.5 dB. The conversion spectrum from the TE1 to TE2 has a 3-dB bandwidth of 23.0 nm. Modal crosstalks from the TE0 are lower than −23.2 dB. The peak transmission of the mode conversion is −0.2 dB.
Fig. 11 (a) Simulated x-component intensity of electric field at the wavelength of 1,550.0 nm, and (b) modal analyses of the transmission spectra with the TE1 incident in the G02-G01.
4. Experimental results
The proposed structure was fabricated on a silicon-on-insulator (SOI) platform. The mode generation and detection are realized by using the asymmetric directional coupler (ADC) [41]. Three TE modes are considered and the ADCs for conversion between TE0 and TE1 (ADC01) as well as that between TE0 and TE2 (ADC02) are applied. Therefore, three input ports on the left and three output ports on the right were designed in our device as shown in Fig. 12(a). Port i can multiplex or demultiplex the mode TEi (i = 0, 1, 2) with the aid of the ADCs. The waveguides and grating couplers were defined by the e-beam lithography (EBL) and fabricated by the fully-etching of the silicon layer. Figure 12(b) and (c) show the scanning electron micrographs (SEM) of the ADC01, where the width of the two waveguides are 860 nm and 392 nm, respectively. The gap width between the waveguides was measured as 191 nm. The grating coupler was realized by fully-etched photonic crystal structures as shown in Fig. 12(d) [42].
Fig. 12 SEM images of (a) the proposed whole device, (b) the ADC01, (c) the waveguides of the ADC01, and (d) the grating coupler.
In our experiment, we considered the G01 and G02 combined structure to realize the conversion from the TE2 to TE1. The grating was formed on the PMMA film by another step of EBL. After developing and baking, the PMMA film turns to be the expected grating pattern of the G01 and G02, as shown in Fig. 13. The width of the silicon waveguide is 1,200 nm to support the first three order TE modes. The grating period of G01 and G02 are 290.0 nm and 316.0 nm, respectively. The grating structures are designed with apodization by linearly changing the LFF from 0 to 1.0 as illustrated in Fig. 7(c). In this way, the coupling coefficient will change as the half-period sine-shape profile. The grating lengths of G01 and G02 are both 350 μm. The thickness of the PMMA layer is approximately 120 nm, which is considered as the grating depth. Therefore, the coupling coefficient of the G01 and G02 are calculated as 180 cm−1 and 160 cm−1, respectively. When we fabricated the cladded PMMA surface grating, there exists an alignment error up to 90 nm, which is limited by our fabrication condition. The effect of this alignment error will be discussed in Section 5.
Fig. 13 SEM images of the apodized (a-c) G01 and (d-f) G02 by linearly changing the LFF from 0 to 1.0.
Figure 14 shows the schematic diagram of the measurement setup of the proposed device. An amplified spontaneous emission (ASE) light source and an optical spectrum analyzer (OSA) are used to measure the transmission spectrum. A pure TE polarized mode can be ensured by introducing a polarizer and a polarization controller. Two fiber ends are used to couple with the proposed device. The input ASE light is coupled with one of the left three input grating couplers via one fiber end, and the output light is coupled with the right three output grating couplers and then connected with an OSA. The transmitted power from left Port i to right Port j is labeled as Tij.
Fig. 14 Schematic diagram of the measurement setup. ASE, amplified spontaneous emission; Pol, Polarizer; PC, Polarization controller; OSA, Optical spectrum analyzer.
We first measured the spectra of T11 and T00 as shown in Fig. 15(a). A notch near 1,525 nm was observed in the T11 curve, which corresponds to the reflection by the G01. The 3-dB bandwidth of the notch is approximately 6.9 nm. A deeper notch is observed in the T00 curve, which is due to the reflections by both G01 and G02. The 3-dB bandwidth of the notch is approximately 9.0 nm. Therefore, the conversion wavelengths of G01 and G02 are very close. Here we found small ripples in the T00 and T11 curves, which is because of the imperfection of the ADC and is explained below.
The spectra of T22 and T21 was also measured. As shown in Fig. 15(b), a notch is also observed in the T22 curve and is caused by the reflection of G02. A passband with a 3-dB bandwidth of 8.3 nm is observed in the T21 curve near 1,525 nm. In the passband, the input TE2 was reflected by the G02 to a TE0. Then, the TE0 is further reflected by G01 and converted to a TE1 which was collected at right Port 1.
Small ripples are observed in both T00, T11, T22, and T21 curves. We measured the spectra of T21 where the PMMA gratings were removed. As shown in Fig. 16(a), crosstalks between the T21 with and without gratings are up to −7.0 dB and is caused by the imperfection of ADC. Therefore, we thought one of the possibilities that cause the ripples is the imperfect fabrication of the ADC.
Fig. 16 (a) Measured spectra of the T21 with and without gratings. (b) Schematic diagram of the optical paths of two TE1s.
The ADC02 is designed to couple the TE0 to TE2, while some residual conversion from the TE0 to TE1 also exists due to the fabrication error. Therefore, the TE1 converted by G01 and G02 and that induced by imperfect ADC02 will propagate together and cause interference due to different optical paths, as illustrated in Fig. 16(b). The phase difference is written as
where βi is the propagation constant of TEi mode (i = 0, 1, 2). φ01 and φ02 are the reflection phase changes of the G01 and G02, respectively. Wavelengths of the resonation notches satisfy the condition: Δφ(λ) = 2mπ + π, where m is an integer. Then, the notch period of the ripple is calculated to be 0.625 nm which agrees well with the measured notch period of approximately 0.610 nm.5. Discussion
5.1 Special mode converters
Generally, various mode conversion spectral shapes can be achieved by the proposed structures according to the theories of the BGs. For example, we have designed a converter with a narrowband transmission peak where mode conversion doesn’t occur, while the mode conversion occurs near the peak.
We combine a uniformed G12 and a π-phase-shifted G02 as shown in Fig. 17(a). The π-phase-shifted G02 provides a narrowband transmission peak without reflection. Therefore, at the wavelengths around the peak, the incident TE0 is reflected by the G02 and then converted to a TE2. Afterward, the TE2 is reflected as a TE1 by the G12 and then passes through the G02.
Fig. 17 (a) Schematic of the proposed converter with a uniform G12 and a π-phase-shifted G02. Calculated spectra of the transmission of (b) the TE0 and TE1 with incident TE0.
The mode conversion spectra from the TE0 to TE1 are obtained in Fig. 17(b) which are calculated by the MTMM. A notch is observed in the middle of the conversion spectra, which corresponds to a narrowband transmission peak of the TE0. The narrowband transmission peak can be also realized by the planar moire grating, which consists of two adjacent uniform gratings located in two transverse sides of the waveguide [40]. However, the moire grating only consider the TE0 mode, where the input and output mode are both TE0 modes. In our structures, mode conversions from TE2 to TE1 occurs at two sides out of the transmission peak.
5.2 Fabrication tolerance analysis
The proposed structure can be fabricated by EBL or stepper photolithography. Fabrication errors typically include the waveguides parameters (i.e. width and height), and the alignment errors during exposure, which are calculated and presented in Fig. 18. The parameters in these calculations are consistent with the experimental structures: waveguide width w = 1,200.0 nm, waveguide height h = 220.0 nm, grating depth d = 120.0 nm in the PMMA layer.
Fig. 18 Conversion wavelength shifts caused by the error of waveguide (a) width and (b) height. (c) Coupling coefficients and (d) full bandwidths variations with respect to alignment errors. The inset in (d) illustrates the schematic of the alignment error in G01.
The conversion wavelength shifts of G01, G02, and G12 are calculated with the waveguide width deviations within ± 40.0 nm, which is caused by the proximity effect during the e-beam exposure [43]. As shown in Fig. 18(a), when the width is varied from −40.0 to 40.0nm, the wavelength shift of G12 is the largest ranging from −22.0 to 20.0 nm and that of G01 is the smallest ranging from −7.1 to 6.4 nm.
The conversion wavelength shifts of G01, G02, and G12 are calculated with the waveguide height deviations within ± 10.0 nm, which is due to the thickness error of the SOI wafers. As shown in Fig. 18(b), when the height is varied from −10.0 to 10.0nm, the wavelength-shift range of these gratings are both approximately ± 22.0 nm.
Alignment error is one of the typical fabrication errors in photolithography processes. In our designs, the coupling coefficients and full bandwidths of the gratings are affected by the alignment errors as shown in Figs. 18(c)-(d). When the alignment error is varied from 0 to 100.0 nm, the coupling coefficient of G01 degrades from 180.2 to 162.0 cm−1, and that of G02 and G12 both degrades from 160.1 to approximately 127.0 cm−1. As a consequence, the full bandwidth of G01 degrades from 7.02 to 6.85 nm, that of G02 degrades from 6.34 to 6.08 nm, and that of G12 degrades from 6.12 to 5.90 nm.
6. Conclusions
In summary, we proposed a forward mode converter based on two cascaded Bragg gratings. The Bragg gratings were designed to obtain a high-efficient coupling between different spatial modes. The proposed structures were simulated via the rigorous 3D-FDTD method, and the mode conversion was verified by the simulation results. Futhermore, we also studied the proposed structures by experiment. Mode conversion was observed from the experimental results. Therefore, the proposed structure may provide a new way for multifunctional mode converters.
Funding
Chinese National Key Basic Research Special Fund (2017YFA0206401); Jiangsu Science and Technology Project (BE2017003-2); National Natural Science Foundation of China (NSFC) (61435014, 11574141, 61504170, 61504058); Natural Science Foundation of Jiangsu Province (BK20160907).
References
1. J. Sun, E. Timurdogan, A. Yaacobi, Z. Su, E. S. Hosseini, D. B. Cole, and M. R. Watts, “Large-scale silicon photonic circuits for optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 20(4), 264–278 (2014). [CrossRef]
2. D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: Polarization handling, light non-reciprocity and loss reduction,” Light Sci. Appl. 1(3), e1 (2012). [CrossRef]
3. D. Dai, J. Wang, and Y. Shi, “Silicon mode (de)multiplexer enabling high capacity photonic networks-on-chip with a single-wavelength-carrier light,” Opt. Lett. 38(9), 1422–1424 (2013). [CrossRef] [PubMed]
4. P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express 17(26), 24342–24348 (2009). [CrossRef] [PubMed]
5. V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012). [CrossRef] [PubMed]
6. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011). [CrossRef]
7. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef] [PubMed]
8. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003). [CrossRef] [PubMed]
9. D. Chen, X. Xiao, L. Wang, Y. Yu, W. Liu, and Q. Yang, “Low-loss and fabrication tolerant silicon mode-order converters based on novel compact tapers,” Opt. Express 23(9), 11152–11159 (2015). [CrossRef] [PubMed]
10. D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express 20(12), 13425–13439 (2012). [CrossRef] [PubMed]
11. D. Dai and H. Wu, “Realization of a compact polarization splitter-rotator on silicon,” Opt. Lett. 41(10), 2346–2349 (2016). [CrossRef] [PubMed]
12. T. Shimizu, N. Hatori, Y. Urino, and T. Yamamoto, “Hybrid Integration Technology of Laser Source with Platform by Flip-chip Bon ding for Silicon Photonics,” IEEE Int. Conf. Gr. IV Photonics GFP 6–9 (2013).
13. N. Riesen and J. D. Love, “Design of mode-sorting asymmetric Y-junctions,” Appl. Opt. 51(15), 2778–2783 (2012). [CrossRef] [PubMed]
14. J. B. Driscoll, R. R. Grote, B. Souhan, J. I. Dadap, M. Lu, and R. M. Osgood, “Asymmetric Y junctions in silicon waveguides for on-chip mode-division multiplexing,” Opt. Lett. 38(11), 1854–1856 (2013). [CrossRef] [PubMed]
15. W. Chen, P. Wang, T. Yang, G. Wang, T. Dai, Y. Zhang, L. Zhou, X. Jiang, and J. Yang, “Silicon three-mode (de)multiplexer based on cascaded asymmetric Y junctions,” Opt. Lett. 41(12), 2851–2854 (2016). [CrossRef] [PubMed]
16. Y. Huang, G. Xu, and S. T. Ho, “An ultracompact optical mode order converter,” IEEE Photonics Technol. Lett. 18(21), 2281–2283 (2006). [CrossRef]
17. N. Erim, I. H. Giden, M. Turduev, and H. Kurt, “Efficient mode-order conversion using a photonic crystal structure with low symmetry,” J. Opt. Soc. Am. B 30(11), 3086–3094 (2013). [CrossRef]
18. L. H. Frandsen, Y. Elesin, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide,” 2013 IEEE Photonics Conf. IPC 201322(7), 333–334 (2013). [CrossRef]
19. D. Ohana and U. Levy, “Mode conversion based on dielectric metamaterial in silicon,” Opt. Express 22(22), 27617–27631 (2014). [CrossRef] [PubMed]
20. M. Ono, H. Taniyama, H. Xu, M. Tsunekawa, E. Kuramochi, K. Nozaki, and M. Notomi, “Deep-subwavelength plasmonic mode converter with large size reduction for Si-wire waveguide,” Optica 3(9), 999–1005 (2016). [CrossRef]
21. P. Wahl, T. Tanemura, N. Vermeulen, J. Van Erps, D. A. B. Miller, and H. Thienpont, “Design of large scale plasmonic nanoslit arrays for arbitrary mode conversion and demultiplexing,” Opt. Express 22(1), 646–660 (2014). [CrossRef] [PubMed]
22. D. Zhu, J. Zhang, H. Ye, Z. Yu, and Y. Liu, “Design of a broadband reciprocal optical diode in multimode silicon waveguide by partial depth etching,” Opt. Commun. 418(March), 88–92 (2018). [CrossRef]
23. J. Kim, S.-Y. Lee, Y. Lee, H. Kim, and B. Lee, “Tunable asymmetric mode conversion using the dark-mode of three-mode waveguide system,” Opt. Express 22(23), 28683–28696 (2014). [CrossRef] [PubMed]
24. W. Jin and K. S. Chiang, “Mode switch based on electro-optic long-period waveguide grating in lithium niobate,” Opt. Lett. 40(2), 237–240 (2015). [CrossRef] [PubMed]
25. Y. B. Cho, B. K. Yang, J. H. Lee, J. B. Yoon, and S. Y. Shin, “Silicon photonic wire filter using asymmetric sidewall long-period waveguide grating in a two-mode waveguide,” IEEE Photonics Technol. Lett. 20(7), 520–522 (2008). [CrossRef]
26. H. Qiu, J. Jiang, P. Yu, T. Dai, J. Yang, H. Yu, and X. Jiang, “Silicon band-rejection and band-pass filter based on asymmetric Bragg sidewall gratings in a multimode waveguide,” Opt. Lett. 41(11), 2450–2453 (2016). [CrossRef] [PubMed]
27. J. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13(11), 4180–4184 (2005). [CrossRef] [PubMed]
28. J. M. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Optical add-drop multiplexers based on the antisymmetric waveguide Bragg grating,” Appl. Opt. 45(6), 1236–1243 (2006). [CrossRef] [PubMed]
29. C. R. Giles, “Lightwave applications of fiber bragg gratings,” J. Lightwave Technol. 15(8), 1391–1404 (1997). [CrossRef]
30. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]
31. M. Li, L. Y. Shao, J. Albert, and J. Yao, “Tilted fiber bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011). [CrossRef]
32. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013). [CrossRef] [PubMed]
33. Y. Dai, X. Chen, J. Sun, Y. Yao, and S. Xie, “Dispersion compensation based on sampled fiber bragg gratings fabricated with reconstruction equivalent-chirp method,” IEEE Photonics Technol. Lett. 18(8), 941–943 (2006). [CrossRef]
34. M. Denni, Sullivian, Electromagnetic Simulation Using the Fdtd Method (Wiley, 2013).
35. X. Wang, “Silicon Photonic Waveguide Bragg Gratings,” PhD Thesis, Univ. Br. Columbia (2013).
36. Q. Liu, Z. Gu, J. S. Kee, and M. K. Park, “Silicon waveguide filter based on cladding modulated anti-symmetric long-period grating,” Opt. Express 22(24), 29954–29963 (2014). [CrossRef] [PubMed]
37. R. R. Grote, J. B. Driscoll, C. G. Biris, N. C. Panoiu, and R. M. Osgood Jr., “Weakly modulated silicon-dioxide-cladding gratings for silicon waveguide Fabry-Pérot cavities,” Opt. Express 19(27), 26406–26415 (2011). [CrossRef] [PubMed]
38. H. Yun, Z. Chen, Y. Wang, J. Fluekiger, M. Caverley, L. Chrostowski, and N. A. F. Jaeger, “Polarization-rotating, Bragg-grating filters on silicon-on-insulator strip waveguides using asymmetric periodic corner corrugations,” Opt. Lett. 40(23), 5578–5581 (2015). [CrossRef] [PubMed]
39. D. T. Spencer, M. Davenport, S. Srinivasan, J. Khurgin, P. A. Morton, and J. E. Bowers, “Low kappa, narrow bandwidth Si(3)N(4) Bragg gratings,” Opt. Express 23(23), 30329–30336 (2015). [CrossRef] [PubMed]
40. S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide moiré grating,” Opt. Express 25(21), 24960–24973 (2017). [CrossRef] [PubMed]
41. B. A. Dorin and W. N. Ye, “Two-mode division multiplexing in a silicon-on-insulator ring resonator,” Opt. Express 22(4), 4547–4558 (2014). [CrossRef] [PubMed]
42. L. Liu, M. Pu, K. Yvind, and J. M. Hvam, “High-efficiency, large-bandwidth silicon-on-insulator grating coupler based on a fully-etched photonic crystal structure,” Appl. Phys. Lett. 96(5), 051125 (2010). [PubMed]
43. C. Vieu, F. Carcenac, A. Pépin, Y. Chen, M. Mejias, A. Lebib, L. Manin-Ferlazzo, L. Couraud, and H. Launois, “Electron beam lithography: Resolution limits and applications,” Appl. Surf. Sci. 164(1–4), 111–117 (2000). [CrossRef]
