2.1.

Structure, Principle, and Design

Figure 1 shows the schematic configuration of the proposed digital DC consisting of $Q$ stages of arrayed bidirectional CMWGs, which are switched by multiple MZSs. The bidirectional CMWGs introduced here can provide positive or negative dispersion, depending on the direction of light propagating in them. For ease of scalability, all the bidirectional CMWGs are designed identically with the same dispersion, denoted as $D_0$, while the sign of dispersion unit $D_0$ can be switched by selecting the input port. Here a $1\times 2$ optical switch is placed at the input side for switching the input light to the target port of a bidirectional CMWG, in which way positive or negative dispersion is realized freely. Another $2\times 1$ optical switch is added at the output side to route the light propagating bidirectionally to the same output port. Hence, the fixed dispersion step for any bidirectional CMWG is given as $\alpha D_0$, where $\alpha =+1$ for positive dispersion and $\alpha =-1$ for negative dispersion.

Fig. 1

Schematic diagram of the proposed digital DC consisting of $Q$ stages of arrayed bidirectional CMWGs which are switched by MZSs.

For the proposed digitally tunable DC, there are $p$ identical bidirectional CMWGs in cascade at the $q$’th stage, and the number $p$ is given by

As a result, the total number of the CMWGs included in the DC is given as $2^Q-1$. To achieve digital dispersion-tuning, $2\times 2$ thermo-optic MZSs with low random phase errors, as proposed in our previous work,³⁸ are utilized at each stage. By switching the MZSs appropriately, the number $P$ of the CMWGs involved for the dispersion accumulation is given by

With this structure, the dispersion can be digitally tuned from $-(2^Q-1)$ $D_0$ to $(2^Q-1)$ $D_0$ with a tuning step of $D_0$, enabling a large dispersion tuning range of $2(2^Q-1)D_0$.

As a key component of the DC, the CMWG is designed with asymmetric grating corrugations, which enables the conversion from the forward ${\mathrm{TE}}_0$ mode to the reflected ${\mathrm{TE}}_1$ mode backward according to the coupled-mode theory.³⁹ Particularly, the grating is chirped by gradually varying the waveguide widths along the lengthwise direction to make the light reflected at different positions depending on the operation wavelength. Since the effective indices of the ${\mathrm{TE}}_0$ and ${\mathrm{TE}}_1$ modes in the waveguide are proportional to the waveguide core width, one has a shorter-wavelength light reflection at the position with a narrower core width. As a result, when light is input from the port with a narrow core, one has positive dispersion [see Fig. 2(a)]. Otherwise, when light is input from the other port with a wide core, one has negative dispersion [see Fig. 2(b)]. The reflected ${\mathrm{TE}}_1$ mode is then dropped to the ${\mathrm{TE}}_0$ mode at the drop port with high efficiency and low crosstalk by utilizing a ${\mathrm{TE}}_0/{\mathrm{TE}}_1$ mode (de)multiplexer. This combination of the bidirectional CMWG and the mode (de)multiplexers eliminates the need for optical circulators, whose monolithic integration is still very challenging.⁴⁰ In this paper, the mode (de)multiplexers are designed with an adiabatic dual-core taper waveguide⁴¹ whose widths are chosen optimally according to the dispersion curves of silicon photonic waveguides with a 220-nm-thick top-silicon layer ($n_{\mathrm{si}}=3.455$), a buffer silica cladding ($n_{\mathrm{sio}2}=1.445$), and an upper silica cladding.

Fig. 2

The present bidirectional CMWG: (a) the operation with positive dispersion ($\alpha =+1$); (b) the operation with negative dispersion ($\alpha =-1$); and (c) structural parameters.

The conversion from the forward ${\mathrm{TE}}_0$ mode to the backward ${\mathrm{TE}}_1$ mode is realized by choosing the grating period according to the Bragg condition:

Figures 3(a) and 3(b) show the calculated results of the transmission spectral responses and the group delays using the transfer matrix method.⁴⁴ Here the propagation loss is reasonably assumed to be $1\mathrm{dB}/\mathrm{cm}$ according to the experimental results, and the apodization length ratio $\beta$ is chosen as 0, ${1/2}^7$, ${1/2}^5$, ${1/2}^3$, and $1/2$, respectively. As shown in Fig. 3(a), the bandwidth is shrunken slightly from 25 to 18 nm when $\beta$ increases from 0 to 1/2, and the transmissions within the bandwidth are kept to be low loss. From Fig. 3(b), it can be seen that there are notable ripples as large as $\pm 15\mathrm{ps}$ when $\beta =0$ (i.e., no apodization is introduced), while the ripples are well suppressed as $\beta$ increases. More details in the wavelength range of 1540 to 1545 nm are given in Figs. 3(c) and 3(d). When $\beta$ increases to be more than ${1/2}^5$, the ripples become almost zero in the wavelength band of 1540 to 1545 nm, as shown in Fig. 3(d). On the other hand, it is often desired to achieve the group delay varied linearly as the wavelength increases. From Fig. 3(d), it can be seen that the linearity becomes worse when the grating is fully apodized (i.e., $\beta =1/2$).⁴⁵ Figure 3(e) shows the calculated average GDR as $\beta$ increases from 0 to 1/2. Here the average GDR is defined as the averaged deviation between the simulated value of the group delay and the target value fitted linearly.⁴⁶ As it can be seen, one has a GDR of $\pm 13.63$, $\pm 4.94$, $\pm 1.42$, $\pm 1.27$, and $\pm 4.37\mathrm{ps}$, respectively, when $\beta =0$, ${1/2}^7$, ${1/2}^5$, ${1/2}^3$, and 1/2. Here we choose $\beta =0.05$ for achieving low ripples of $\pm 1.18\mathrm{ps}$ as well as high linearity in the wavelength range 1532 to 1548 nm. As a result, for the designed bidirectional CMWG, the 3-dB bandwidth is about 23.7 nm, the maximal group delay is 115 ps, and the dispersion is about $+6.30\mathrm{ps}/\mathrm{nm}$. On the other hand, a negative dispersion of $-5.86\mathrm{ps}/\mathrm{nm}$ is produced when light is input from the other end with wide core.

Fig. 3

Simulated responses for the CMWG with different apodization length ratios $\beta$. (a) Transmission spectrum; (b) group delay spectrum; (c) group delay spectrum at 1540 to 1545 nm; (d) group delay spectra with linear data fitting; and (e) average GDR variation as the apodization length ratio $\beta$ varies.

The digital dispersion tuning is achieved by appropriately switching the MZSs that are designed according to our previous work.³⁸ Here we consider the DC with four stages (i.e., $Q=4$), regarding the limitation due to the loss issue in experiments. According to the structure shown in Fig. 1, there are 15 CMWGs available at maximum. Figures 4(a)–4(d) show the transmission and the group delay of the DC operating with positive or negative dispersion as the total number $P$ of the involved CMWGs increases. When the DC operates with positive dispersion, the longer wavelength experiences a longer propagation distance, and thus a higher excess loss, as shown in Fig. 4(a). In contrast, when the DC operates with negative dispersion, the longer wavelength experiences a shorter propagation distance, and thus a lower excess loss, as shown in Fig. 4(b). Since the maximal propagation distance for the longest wavelength with the maximal positive dispersion is about 12 cm, the longest wavelength shows the maximal delay of 1736 ps and the highest total loss of about 12 dB as the propagation loss is assumed to be $\sim 1\mathrm{dB}/\mathrm{cm}$ here, as shown in Figs. 4(a) and 4(c). Similarly, when operating with negative dispersion, the shortest wavelength has the maximal delay of 1663 ps and the highest total loss of about 12 dB [see Figs. 4(b) and 4(d)]. In this case, the designed DC enables digital dispersion tuning from $-87.90$ to $94.43\mathrm{ps}/\mathrm{nm}$ with a step $D_0$ of 6.30 and $-5.86\mathrm{ps}/\mathrm{nm}$ for positive and negative dispersions, respectively, as shown in Fig. 4(e). Correspondingly, the total dispersion-tuning range is $182.33\mathrm{ps}/\mathrm{nm}$, and the total delay tuning range reaches 3399 ps.

Fig. 4

Simulated results for the DC operating with positive or negative dispersion as the total number $P$ of the involved CMWGs increases. Calculated transmissions for the case with (a) positive or (b) negative dispersions. Calculated group delay for the case with (c) positive or (d) negative dispersions. (e) The dispersion-tuning range as the CMWG number increases.

Note that there is a critical wavelength (${\lambda }_{\mathrm{cP}}$ or ${\lambda }_{\mathrm{cN}}$) in the CMWG passband defined by ${\lambda }_1<\lambda <{\lambda }_2$, where ${\lambda }_1$ and ${\lambda }_2$ are the edge-wavelengths for the passband, as shown in Figs. 4(a)–4(d). For the case of operating with positive dispersion shown in Figs. 4(a) and 4(c), the propagation loss and the group delay increase linearly with the difference $|\lambda -{\lambda }_{\mathrm{cP}}|$ when ${\lambda }_{\mathrm{cP}}<\lambda <{\lambda }_2$. In contrast, when ${\lambda }_1<\lambda <{\lambda }_{\mathrm{cP}}$, the group delay and the propagation loss are kept constant almost, as shown in Figs. 4(a) and 4(c). Similarly, for the case of operating with negative dispersion shown in Figs. 4(b) and 4(d), the propagation loss and the group delay increase linearly with the difference $|\lambda -{\lambda }_{\mathrm{cN}}|$ when ${\lambda }_1<\lambda <{\lambda }_{\mathrm{cN}}$. In contrast, when ${\lambda }_{\mathrm{cN}}<\lambda <{\lambda }_2$, the group delay and the propagation loss are kept constant almost, as shown in Figs. 4(b) and 4(d). This happens for a Bragg grating designed with the core width chirped very slowly.^32,47 Here we give an explanation by considering the case of operating with positive dispersion shown in Figs. 4(a) and 4(c) as an example. For the present CMWG, the front-end part is designed with an appropriate period to reflect the wavelength ${\lambda }_{\mathrm{cP}}$ completely within only a few grating periods. Since the chirping of the CMWG is gentle, the wavelength within the range shorter than ${\lambda }_{\mathrm{cP}}$ can still be reflected very well with low losses and accordingly it has a group delay similar to the wavelength ${\lambda }_{\mathrm{cP}}$. Therefore, one should choose the effective wavelength range as ${\lambda }_{\mathrm{cP}}<\lambda <{\lambda }_2$ for achieving the positive dispersion, and the wavelength range of ${\lambda }_1<\lambda <{\lambda }_{\mathrm{cP}}$ should be ignored. Similarly, the effective wavelength range for achieving negative dispersion is given by ${\lambda }_1<\lambda <{\lambda }_{\mathrm{cN}}$ and the wavelength range of ${\lambda }_{\mathrm{cN}}<\lambda <{\lambda }_2$ should be ignored.

2.2.

Experimental Results

Figure 5 shows the optical microscope image of the fabricated DC, including the zoom-in view of the input couplers, the MZSs, the region for positive dispersion-tuning, and the region for negative dispersion-tuning. The total footprint of the present DC is about $5\mathrm{mm}\times 0.38\mathrm{mm}$. Particularly, the long straight waveguides for connections on the chip are widened to lower the propagation loss greatly.⁴⁸ For the present DC, there are 12 MZSs in total for the digital dispersion tuning. Among them, MZSs S02 to S06 and MZSs S08 to S12 are used for the cases with positive and negative dispersions, respectively. Meanwhile, MZSs S01 and S07 are used to switch at the input/output ports to achieve positive/negative dispersion.

Fig. 5

Microscope images of the fabricated DC: (a) full view of the DC; zoom-in view of the (b) input coupler, (c) the MZS, (d) the region for positive dispersion-tuning, and (e) the region for negative dispersion-tuning. GC, grating coupler; EC, edge coupler.

Figure 6(a) shows the measured transmission spectrum of a pair of mode (de)multiplexers, and Fig. 6(b) shows the measured transmission spectrum of the cross/bar ports of an MZS at the OFF/ON states. These results are normalized with the transmission of a straight single-mode silicon photonic waveguide fabricated on the same chip. From Fig. 6(a), it can be seen that the fabricated mode (de)multiplexer has an excess loss of $<1\mathrm{dB}$ and intermode crosstalk of $<-24\mathrm{dB}$ in the wavelength range of 1525 to 1560 nm, as expected in theory. It is crucial to achieve high extinction ratios for MZS used in system so that the crosstalk can be minimized. Fortunately, the measurement results given in Fig. 6(b) show that the MZS works very well with high performances, exhibiting a low excess loss of $<1\mathrm{dB}$ and a high extinction ratio of $>28\mathrm{dB}$ for both OFF and ON states. In addition, the applied voltage is $\sim 2.75\mathrm{V}$ when operating at the ON state, and the thermal-optic switching speed is around microseconds, which is similar to the results for silicon photonic devices reported previously.⁴⁹ Figure 6(c) shows the measured spectral responses for a single CMWG ($P=1$) as the temperature varies from 27°C to 87°C. It can be seen that the center wavelength is red-shifted linearly with a slope of $77.6\mathrm{pm}/°\mathrm{C}$, and the measured group-delay response has the same redshifts as well. As a result, the working temperature should be stabilized to avoid any random variation. On the other hand, it is possible to achieve fine tuning of the group-delay thermally. Figures 6(d) and 6(e), respectively, show the measured transmissions of the present DC operating with positive or negative dispersions when different numbers of CMWGs are involved. Here the CMWG number $P$ is given as 1, 3, 5, 7, 9, 11, 13, and 15. From the measurement results, the average excess losses for each CMWG are 0.64 and 1.02 dB within the wavelength bandwidth of 20 nm for positive and negative dispersion, respectively. When all CMWGs are involved (i.e., $P=15$), the measured propagation loss varies from 9.39 dB at 1530.2 nm to 25.28 dB at 1546.9 nm for the case with positive dispersion and from 23.48 dB at 1530.7 nm to 15.21 dB at 1547 nm for the case with negative dispersion, respectively. The difference between the two cases with positive and negative dispersions is due to that the mode (de)multiplexers have larger excess losses at longer wavelengths [shown in Fig. 6(a)]. The power fluctuations at 1531.5 and 1544 nm are mainly due to the field stitching of E-beam writing and related fabrication errors, which can be lowered greatly if the fabrication processes are improved. Moreover, the fluctuation accumulates as the number of the CMWGs in cascade increases.

Fig. 6

Measured transmissions of the fabricated devices. (a) Measured transmission for a pair of mode (de)multiplexers. (b) Measured transmissions at the cross/bar ports of the MZS at the OFF/ON states. (c) Measured spectral responses when selecting a single CMWG ($P=1$) at different temperatures. Measured transmission of the present DC operating with (d) positive dispersion and (e) negative dispersion when different numbers of CMWGs are involved.

Figure 7(a) shows the measured group-delay spectrum of the present DC (see the data labeled by the circles), providing different group delays from the negative value ($\alpha =-1$) to the positive one ($\alpha =+1$) by switching the MZSs at the input/output ports and the MZSs for the digital tuning. Here the CMWG number $P$ is given as 1, 3, 5, 7, 9, 11, 13, and 15. It can be seen that the group delay increases or decreases linearly as the wavelength increases in the wavelength band of 1532 to 1549 nm. In contrast, the group delay in the wavelength range of 1529 to 1532 nm is insensitive to the wavelength, which is consistent with the theoretical results shown in Fig. 4(c) (see the wavelength range of 1526 to 1532 nm). The measured data in Fig. 7(a) are also fitted linearly, as shown by the dashed lines. Figure 7(b) shows the corresponding positive/negative dispersions as the number of the CMWGs involved increases. The dispersion is digitally tuned from $-61.53$ to $63.77\mathrm{ps}/\mathrm{nm}$ with a step of $4.16\mathrm{ps}/\mathrm{nm}$, which is produced by each CMWG. When all CMWGs are involved (i.e., $P=15$) by turning on all the $2\times 2$ MZSs, the group delay can be varied from $-1021$ to 1037 ps when switching the $1\times 2$ MZS at the input side and $2\times 1$ MZS at the output side, enabling a total group-delay tuning range of 2058 ps. The measured waveforms shown in Figs. 7(c) and 7(d) clearly demonstrate the measured group delay of the DC when operating at different wavelengths of 1532, 1534, …, 1546 nm. The delay loss of the fabricated DC is estimated as 15.32 and $8.10\mathrm{dB}/\mathrm{ns}$ for the operations with positive and negative dispersions, respectively.

Fig. 7

Measured group-delay and dispersion of the fabricated DC. (a) Measured (circles) and linearly fitted (dashed lines) group-delay by switching the $1\times 2$ and $2\times 1$ MZSs at the input/output ports and all the $2\times 2$ MZSs. (b) Measured dispersion as the number of the CMWGs involved increases. Measured group delay of the DC operating with (c) positive dispersion and (d) negative dispersion at different wavelengths for the case of $P=15$.

2.3.

Comparisons

Table 1 gives a summary of the on-chip DCs reported in recent years. It shows that there are very few on-chip DCs enabling positive/negative dispersion. As demonstrated, the DC with continuously adjustable dispersion was realized using cascaded MZIs,²⁴ but it suffers from large footprints and a narrow bandwidth of $<0.1\mathrm{nm}$. In contrast, the present DC has a compact footprint and is the unique one enabling a high positive/negative dispersion within a broad bandwidth of $\sim 20\mathrm{nm}$. Furthermore, the present on-chip DC is circulator-free and is flexible to be integrated monolithically with other components, which is very useful for many applications.

Table 1

Summary of reported on-chip DCs.

4.1.

Fabrication

The designed DC was fabricated using a standard process for silicon photonics. The top-silicon layer is 220 nm, and the buried oxide thickness is $3\mu \mathrm{m}$. A thin-film silica upper cladding was deposited on the top, followed by overlaying a titanium nitride metal heater for thermo-optic MZSs. There is a deep air trench between any two phase-shifters for thermal isolation. The microheaters on the MZSs are connected to the electrical pads via high-conductivity metal lines, and all these electrical pads are wire-bonded to a printed circuit board.

4.2.

Experimental Setup

To measure the transmission spectra of the devices on the chip, the setup with a broad-band amplified spontaneous emission light source and an optical spectrum analyzer was used. Moreover, Fig. 8 shows the experimental setup for measuring the group delay and the dispersion. With this setup, light emitted from a tunable laser source (TLS) first goes through a polarization controller (PC) and then an electro-optic (EO) modulator. Here the EO modulator is operated with a 5 to 40 GHz radio frequency signal from a synthesized clock generator. Consequently, the modulated light was amplified with an erbium-doped fiber amplifier (EDFA) and finally was coupled to the on-chip DC. The signal output from the chip was then detected using a digital communication analyzer (DCA). In the experiment, a multichannel voltage source was employed for the operations of MZSs.

Fig. 8

Experimental setup for the measurement of the group delay and the dispersion. TLS, tunable laser source; PC, polarization controller; IM, intensity modulator; EDFA, erbium-doped fiber amplifier; SCG, synthesized clock generator; and DCA, digital communication analyzer.