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Abstract
Highly linear electro-optic modulators are key components in analog microwave photonic links, offering on-chip direct mixing of optical and RF fields. In this work, we demonstrate a monolithic integrated Michelson interferometer modulator on thin-film lithium niobate (LN), that achieves linearized performance by modulating Bragg grating reflectors placed at the end of Michelson arms. The modulator utilizes spiral-shaped waveguide Bragg gratings on Z-cut LN with top and bottom electrodes to realize extensive reflectors, essential for linearized performance, in a highly integrated form. Optical waveguides are realized using rib etching of LN with precisely engineered bottom and top cladding layers made of silicon dioxide and SU-8 polymer, respectively. The compact design fits a 3 mm long grating in an 80 µm × 80 µm area, achieving a broad operating bandwidth up to 18 GHz. A spurious free dynamic range (SFDR) of 101.2 dB·Hz2/3 is demonstrated at 1 GHz, compared to 91.5 dB·Hz2/3 for a reference Mach-Zehnder modulator fabricated on the same chip. Further enhancement in SFDR could be achieved by reducing fiber-to-chip coupling loss. The proposed demonstration could significantly improve the linearity of analog modulator-based integrated optical links.
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1. Introduction
Microwave photonics (MWP) is an emerging technology that aims to harness the ultra-high capacity and low loss propagation of optical systems to achieve efficient microwave signal generation and high-speed transmission [1]. Among different components, electro-optic (EO) modulators are considered the most attractive elements in MWP systems, being responsible for transforming signals from the radio frequency (RF) domain to the optical domain. Tremendous efforts have been devoted to developing cost-effective and miniaturized chip-scale modulators, capitalizing on recent advances in photonic integrated circuit (PIC) technology [2,3]. Several platforms, such as silicon-on-insulator (SOI) [4], silicon nitride (SiN) [5], indium phosphide (InP) [6], and lithium niobate (LiNbO3, LN) [7], were employed to achieve high-performance modulators with wide bandwidths, low power consumption, and low optical insertion loss. While these parameters are the key performance metrics of EO modulators in the context of digital communications, linearity becomes the most crucial specification in analog transmission systems. Linearity is usually characterized using spurious free dynamic range (SFDR), which depends on the linear performance of the modulator as well as insertion loss and system noise [7,8].
Modulator linearity is mainly limited by the sinusoidal transfer function of conventional Mach-Zehnder modulators (MZMs) and the inherent characteristics of the modulation scheme. Different practices have been proposed in recent decades to compensate for the MZM response [9–18]. Such techniques were primarily implemented on the SOI platform, being the leading technology for PICs due to its compatibility with CMOS foundries [19]. However, silicon modulators still showed limited linearity as they mainly depend on the free carrier plasma dispersion effect, which is inherently nonlinear. On the other hand, lithium niobate has been an excellent candidate for linear modulators due to its strong linear EO effect, high refractive index, and wide transparency region [20]. However, devices based on bulk lithium niobate had large footprints and thus poor integration density compared to those on the SOI platform. Recently, thin film lithium niobate (TFLN) technology has emerged and been widely adopted in modern integrated MWP systems, offering the excellent material properties of bulk LN along with strong confinement of optical modes and high potential integration capabilities [7]. While ultra-wideband modulators with low driving voltages have been demonstrated on TFLN [21–24], the measured SFDR of these devices is around ∼100 dB·Hz2/3 [25,26]. Such performance, limited by MZM response, is still far from their bulk counterparts ($> $ 120 dB·Hz2/3 [27]). Therefore, advanced linearization schemes are sought after in the promising platform.
Linearization techniques are based on either electrical [9,10] or optical schemes [11–18]. Electrical methods, such as predistortion linearization [9] and digital postprocessing [10], have the disadvantages of high complexity, high power consumption and limited bandwidth. Therefore, optical methods have received greater interest, including dual parallel or series cascaded modulators, ring-assisted Mach-Zehnder modulator (RAMZM), and grating-assisted Michelson interferometer modulator (GAMIM). Dual parallel and series modulators could achieve SFDR up to 123 dB·Hz6/7 and 109.5 dB·Hz2/3, respectively, on silicon by manipulating the splitting ratio between different modulators [11,12]. The dual parallel MZM (DP-MZM) was recently implemented on LN, demonstrating an SFDR of 110.7 dB·Hz2/3 [13]. Such a configuration has the drawbacks of complicated compensation settings, high insertion loss, and the high cost of multiple modulators. RAMZM relies on the ring resonator's super-linear phase response to balance the MZM's sub-linear response, and has been widely adopted in many research reports. RAMZM has been successfully demonstrated on silicon with SFDR of 111.3 dB·Hz2/3 and on heterogenous III-V/Si platform with SFDR of 117.5 dB·Hz2/3 [14,15]. Moreover, 120.04 dB·Hz4/5 SFDR was measured recently on TFLN [16]. However, this implementation has low RF bandwidth, limited by the cavity photon lifetime, and tight fabrication tolerance, due its naroow optical bandwidth [17]. GAMIM is an alternative structure that uses Bragg grating reflectors (BGRs) at the end of Michelson arms to compensate the sinusoidal MZM dependence by the grating phase response, which approaches inverse cosine for very long gratings [18]. The idea should not be confused with Bragg grating resonator-assisted MZM, which uses the resonator as a phase modulator in MZM arms instead of a ring resonator in RAMZM structure [28]. The drawback of the GAMIM structure is the need for long gratings, which increases the device footprint and limits the modulation bandwidth. It is worth noting that comparing linearization schemes based on reported SFDR values is challenging, as different measurement setups with different signal-to-noise ratios (SNRs) are used in different reports.
In this paper, we implement the GAMIM structure on TFLN, where spiral-shaped waveguide Bragg gratings (WBGs) are used to achieve extensive reflectors in an ultra-compact form. The spiral WBGs were demonstrated as optical filtering devices on SOI and other platforms [29–31], offering the advantages of small footprint and fabrication uniformity over straight WBGs. However, such spiral-shaped devices were not implemented on TFLN, because current WBG designs on LN were achieved on the most commonly used X/Y cut with planar electrodes on both sides of waveguides. Several uniform and phase shifted (PS) WBGs were recently reported using this X/Y cut configuration utilizing LN strong EO effect to modulate WBG's filtering response for tunable filters and EO modulators applications [32–35]. X/Y cut was mainly adopted as it has the advantage of simple fabrication and good optical confinement of optical transverse electric (TE) mode [7,22]. Co-traveling modulation design is essentially employed in EO modulators to achieve phase matching between optical and RF fields for broad bandwidth operation. Nevertheless, X/Y cut results in long devices that are not optimum for compact realization. Recently, a compact MZM was achieved on Z-cut LN based on spiral waveguide phase modulators [36]. Bending could be added to modulated waveguides since they are in-plane isotropic. Electrodes are placed on top and bottom of waveguides and are considered lumped [37]; hence, there is no need for traveling wave modulation design.
The same technology is adopted in this work for GAMIM design. Optical waveguides are realized by partially etching Z-cut LN to get rib waveguides. Silicon dioxide (SiO2) is the bottom cladding, while SU-8 polymer is used as a top cladding to cover the waveguides. Aluminum (Al) and gold (Au) are used as top and bottom metal electrodes, respectively. The design fits a 3 mm long Bragg grating reflector in an 80 µm × 80 µm area. It exhibits an EO tuning sensitivity of 6.2 pm/V and a wide bandwidth operation up to 18 GHz. Single-ended and differential modulation were measured, while SFDR of 101.2 dB·Hz2/3 was demonstrated for intensity modulation using single-ended operation. This represents a 9.7 dB enhancement over a reference MZM structure that was fabricated on the same die with spiral phase modulation waveguides. Based on grating couplers, the device has relatively high fiber-to-chip optical coupling loss and is expected to achieve a higher SFDR if edge coupling is used. The proposed linear modulator has a great potential to significantly improve signal integrity and minimize added spurious harmonics and noise in analog MWP links.
2. Design methodology
The three-dimensional (3D) schematic of the proposed design is presented in Fig. 1(a). Input light is coupled to the modulator using a grating coupler, then split using a 2 × 2 multimode interference (MMI) waveguide into Michelson arms that end with two spiral BGRs. The Michelson interferometer has two unbalanced arms, where length mismatch is used to operate at the quadrature point $({\beta \Delta L = \mathrm{\pi }/2} )$ for optimum linear operation. BGRs are designed with a single reflection port by adding width corrugations to two spiral waveguides that follow an Archimedean trajectory, as shown in Fig. 1(b). The grating structure is mapped to the whole curve, wrapping long uniform WBGs into a highly compact area [29]. A third-order grating with a period of $({\Lambda = 1.2\; \mathrm{\mu m}} )$ was chosen for device characterization as it realized well-shaped gratings with identical responses between both mirrors. The period was selected to achieve a center Bragg wavelength of ${\lambda _B} = 2\overline {{n_{eff}}} \Lambda /m$, where m is the Bragg order and $\overline {{n_{eff}}} $ is the average effective mode index of the waveguide grating, at 1550 nm. A corrugation width of $\Delta w = 200\; \textrm{nm}$ was used around a nominal waveguide width of ${w_g} = 1\; \mathrm{\mu m}$. The inner radius of the circular spiral (${r_o}$) was chosen as 15 µm to balance the tradeoff between effective refractive index variations due to curvature and total BGR footprint. The gap between spiral turns ($g$) was chosen to be 2 µm. The number of grating periods is ${N_G} = 2400$, resulting in a 3 mm long BGR that could be wrapped into a circular spiral with a maximum radius of 40 µm. All components in the optical path were optimized for minimum insertion loss using finite difference time domain (FDTD) simulations.
Fig. 1. (a) 3-D schematic of the proposed GAMIM on Z-cut LN. (b) Top view of the spiral grating mirror schematic showing different design parameters. $\Lambda = 1.2\; \mu m,\; \Delta w = ({{w_2} - {w_1}} )/2 = 200\; nm,\; g = 2\; \mu m,\; {r_o} = 15\; \mu m.$ (c) Cross sectional view of the implemented modulator showing vertical dimensions of different layers. ${n_o}$ and ${n_e}$ are the ordinary and extra-ordinary refractive indices of LN, respectively.
The design of the RF path aims to enhance the EO modulator performance metrics, including modulation efficiency, EO bandwidth, and insertion loss. As the waveguide cross-section is depicted in Fig. 1(c), the stack is composed of a silicon handle layer, a 140 nm thick bottom metallization layer made of Au with chromium (Cr) adhesion layer, a 1 µm silicon dioxide (SiO2) layer, and 800 nm of Z-cut LN. After patterning the LN, a top dielectric is used to cover the devices before adding 200 nm of the top Al metal electrode. SU-8 polymer is utilized as the top cladding, having the advantage of high uniformity, flexible thickness as controlled by spin coating speed, and good optical and dielectric properties $({n_r} \approx 1.57$ at 1550 nm$,\,{\varepsilon _r} \approx 3$) [38–40]. Moreover, it exhibits excellent thermal stability after post exposure baking up to 311°C [41]. The thickness of SU-8, i.e., the distance between the electrode and optical waveguide, is a crucial design parameter that could be manipulated to optimize the modulator performance among different tradeoffs [36]. Higher thickness results in wider bandwidth and lower optical loss, yet at reduced modulation efficiency. A thickness of 1 µm was selected for device implementation.
Linear characteristics of GAMIM are illustrated based on grating reflectivity $(r )$ and its phase response $({\varphi _r}$), which are calculated by [18,42]:
By applying EO modulation to vary the refractive index of the BGRs, the central Bragg frequency will be tuned with rate, $\partial {f_B}/\partial V ={-} {f_B}/\; \overline {{n_{eff}}} \times \partial {n_{eff}}/\partial V$, where $\partial {n_{eff}}/\partial V = n_e^3{r_{33}}\mathrm{\Gamma }/2d,$ is the effective index EO modulation efficiency, ${n_e}$ is the extra-ordinary index of LN, ${r_{33}}$ is the highest LN EO coefficient adopted in Z-cut design, d is the electrode gap, and $\mathrm{\Gamma }$ is the overlap integral between the optical mode and applied electric field. Assuming no frequency offset at zero applied voltage, the detuning change as a function of voltage is given by: $\delta f(V )= $ $\Delta f.V/2{V_\pi }$, where ${V_\pi }$ is defined here as the voltage at which the frequency offset equals half the reflection bandwidth $(\delta f = \Delta f/2$). ${V_\pi }$ can be calculated as:
Hence, the BGR phase response could be expressed as, ${\varphi _r} \approx{-} \textrm{co}{\textrm{s}^{ - 1}}({V/{V_\pi }} ).$ Assuming modulating the GAMIM at the phase quadrature point between both Michelson arms $({\Delta \varphi = \pi /2} )$, the output transfer function for single arm modulation of very long gratings $({\kappa L \to \infty } )$, derived in Appendix A, is given by:
where $P_{out}^{single}$ and ${P_{in}}$ are the optical output and input power, and T is the transmission coefficient of the modulator representing power loss. As depicted, the perfect linear operation could be achieved. The optical transfer function is compared to that of the MZM in Fig. 2(a), at different grating lengths corresponding to $\kappa L$ of 2, 4, and 8, showing the effect of grating length on linearity. Input voltage is normalized to ${V_\pi }$, which is different between MZM and GAMIM cases. Linearity enhancement is better understood using SFDR simulation in Fig. 2(c). Given two input RF signals at frequencies ${f_1}$ and ${f_2}$, the fundamental harmonic (FH) power is compared to third order intermodulation dispersion (IMD3) power at $2{f_1} - {f_2}$ and $2{f_2} - {f_1}$ [7]. Output RF response, which is proportional to the square of optical response, is plotted vs normalized input power, where the input voltage is normalized to ${V_\pi }$. Simulation results shows significant reduction of IMD3 power resulting in SFDR enhancement from 1.2 dB to 27 dB compared to the MZM case, depending on the mirror grating length. The exact SFDR value would change depending on the SNR of the measurements, yet the same SFDR improvement would be expected. A noise floor of -154 dBm/Hz is assumed here, similar to our measurement case at 1 GHz. The main drawback of the device is that the linearity improvement comes at the expense of the fundamental term power, which is lower than the MZM case because the linear region covers the whole range between 0 and ${V_\pi }$ as shown in the optical response. In addition, there is a tradeoff between the grating coupling $\kappa $, which defines the reflection bandwidth $\Delta f$, and the modulator ${V_\pi }$. Another limitation arises from the required operation of the device at the Bragg frequency which makes it sensitive to fabrication tolerances as ${f_B}$ is subject to shift due to change in fabricated grating period or etching depth.Fig. 2. (a), (b) Simulated optical transfer functions of reference MZM and different lengths of GAMIM for (a) single arm intensity modulation (power), and (b) differential amplitude modulation (field). Input voltage is normalized to ${V_\pi }$, for good comparison between MZM and GAMIM. (c), (d) Simulated FH and IMD3 powers vs normalized input RF power of reference MZM and different lengths of GAMIM for (c) single arm intensity modulation (power), and (d) differential amplitude modulation (field). SFDR is indicated for each case, where the noise floor is exported from the measurement setup as -154 dBm/Hz.
Differential modulation would double the device efficiency using push-pull operation. Considering intensity modulation, linearity enhancement compared to MZM is much less than single arm modulation case. That could be illustrated as the phase shift between grating reflectors in both arms would be $\Delta {\varphi _r} \approx{-} 2\textrm{co}{\textrm{s}^{ - 1}}({V/{V_\pi }} )$, which will not compensate for the sinusoidal transfer function of MZM. The optical output power for long gratings, derived in Appendix B, can be given by:
Nevertheless, considering amplitude modulation, differential operation of GAMIM would result in a highly linear output electric field [18]. Assuming $\pi $ phase shift between Michelson arms, the output field, derived in Appendix B, is calculated as:
where ${E_{in}}$ is the optical input field, and t is the field transmission coefficient. The optical field transfer function and SFDR simulation results are shown in Fig. 2(b) and (d), in comparison with the MZM case. A similar SFDR enhancement is observed, ranging from 0.3 dB for short gratings to 28.2 dB for $\kappa L = 8$. Due to differential drive, the FH power is closer to the MZM case in this configuration. The obtained result could be employed in coherent optical links to implement advanced modulation schemes [43–45]. In this work, we consider intensity modulation cases only. However, the modulator allows both single arm and differential modulation, where both cases are experimentally demonstrated.3. Experimental validation
3.1 Fabrication process
The device fabrication procedure implemented for the optical modulator with spiral waveguides is shown in Fig. 3. The initial stack is a Z-cut LN film on top of a buried SiO2 layer and Au bottom electrode, supported by a silicon carrier wafer. First, the LN layer was patterned using electron beam lithography (EBL) after depositing a Cr layer with optimized thickness as a hard mask. Direct etching of 600 nm out of the 800 nm thick LN was employed to minimize propagation loss while supporting optical transverse magnetic (TM) mode. TM mode is chosen to harness the EO effect along the Z-axis of LN. A chlorine-based and a fluorine-based plasma were used to etch Cr and LN films, respectively, resulting in a ∼70° sidewall angle for optical waveguides [36,46]. Next, a second layer of photoresist (PR) was patterned using optical lithography for ground openings to the bottom Au electrode. Then, the SU-8 layer was spin-coated on optical waveguiding structures, except for the grating couplers used for input and output coupling. A hard baking step was performed on SU-8, making it a permanent part of the structure. Finally, the top electrode Al layer was sputtered and patterned above modulated spiral mirrors using the lift-off process. Al was chosen as it was found to minimize waveguide loss relative to other metals at 1550 nm, based on FDTD simulations. Compared to Au, both metals gave almost the same optical loss of 0.2 dB/cm at 1550 nm; however, Au requires adding Cr adhesion layer, which increases the optical loss to 1 dB/cm at 1550 nm.
Fig. 3. Detailed device fabrication process on Z-cut TFLN with top and bottom electrodes. (a) Initial stack. (b) Deposition of Cr hard mask and PR patterning using EBL. (c) Etching Cr and LN. (d) PR patterning and etching LN and SiO2 for ground pad openings. (e) SU-8 spin coating and patterning. (f) PR patterning and Al metal sputtering. (g) Metal lift-off to form the top electrode.
Figure 4 shows the fabricated devices’ optical and scanning electron microscope (SEM) images. The top view microscope image is displayed in Fig. 4(a), showing waveguide bending in both Michelson arms to achieve length mismatch for the unbalanced operation of the interferometer. As seen in the image, long waveguides were used in each Michelson arm to adapt for the positions of electrical and optical probes in the measurement setup. The excess waveguide length may be eliminated when integrating the device with other photonic components resulting in lower propagation loss and denser integration for the modulator. The SEM image of the spiral grating mirror after LN etching is depicted in Fig. 4(b), showing the periods of third-order gratings that were adopted in measurements. A cross-sectional SEM view of the optical waveguide without width corrugations is shown in Fig. 4(c), along with the simulated optical TM mode at 1550 nm.
Fig. 4. (a) Microscope image of the fabricated GAMIM after patterning of top cladding and electrodes. Length mismatch between both arms is achieved using waveguide bending to have the same transmission coefficient. (b) Top-view SEM image of the spiral grating mirror before adding top SU-8 and metal layers, showing the fabricated third order grating in the figure inset. (c) SEM cross-sectional image of one optical waveguide with ∼70° sidewall angle (left) and simulated optical TM mode shape with effective and group refractive indices values (right).
3.2 Measurement results
The fabricated GAMIM device was characterized in detail. First, the transmission spectrum of the unbalanced GAMIM, with a mismatch of $\Delta L = 200\; \textrm{nm}$, was optically measured as shown in Fig. 5(a). A tunable laser source (Santec 710 TSL) and a fiber polarization controller (FPC) were used to adjust the input light's wavelength, power, and polarization to the TM mode. The modulator transfer function is a superposition of the Michelson interferometer response and the grating’s bandwidth-limited reflectivity, resulting in a smooth roll-off at the bandgap edges with an extinction ratio up to 29 dB. The measurement shows the high insertion loss for the device which is dominated by the grating couplers’ loss ($> $ 10 dB each), as well as waveguide scattering loss originating from surface roughness and fabrication nonidealities (measured in our previous work as 1.3 dB/cm [36]). An erbium-doped fiber optical amplifier (EDFA) was used to boost the input signal to 27 dBm to compensate for such optical losses.
Fig. 5. (a) Measured optical response of the unbalanced GAMIM vs wavelength. (b) Transmission spectrum shift of the GAMIM at three different AC voltages, demonstrating EO AC tunability of 6.2 pm/V.
The EO modulation response was characterized by recording the transmission spectra at different voltage levels. The measurement could not be performed at DC due to the inefficient DC response of Z-cut modulators resulting mainly from surface charge distribution effects [7,36,47]. Instead, a 1 kHz square-wave signal, generated from a function generator followed by a 50x piezo amplifier and a power splitter, was applied to both mirrors, where the optical spectrum was measured at different pulse amplitudes [37]. Measured results are shown in Fig. 5(b), demonstrating an EO AC tuning sensitivity of 6.2 pm/V for the central wavelength. Based on ${V_\pi }$ definition for GAMIM in Eq. (5), which is different from the conventional definition for MZM, ${V_\pi }$ can be estimated from the grating reflection bandwidth and the EO tuning sensitivity as 215 V. The experimental value is in good agreement with the theoretically predicted value, where $\partial {n_{eff}}/\partial V$ is calculated from the overlap integral between the applied electric field and optical mode, which is extracted from a COMSOL Multiphysics simulation. To avoid the aforementioned complication of the modulator DC response, the effect of ${V_\pi }$ on the amplitude of the analog-modulated optical signal as a function of the applied voltage is quantitatively analyzed using the following RF measurements.
The RF response of the GAMIM was measured to characterize its linearity at single-ended operation, test the differential operation, and measure the optical modulation amplitude at different applied voltages. Using the two-tone test setup shown in Fig. 6(a), two sinusoidal RF signals, centered at 1 GHz with frequency separation of 10 MHz, were applied to the top electrode of one mirror. The optical power was measured at the output using a high-speed photoreceiver (Thorlabs RXM40AF), followed by an RF spectrum analyzer (Agilent E4445A) which was used to analyze the FH and IMD3 components at various input RF powers. For a one-to-one comparison, a reference MZM was fabricated on the same chip with spiral phase modulation waveguides and a similar electrode design. The laser source wavelength was tuned to obtain maximum linearity at the output. The IMD3 SFDR results for GAMIM and reference MZM are 101.21 dB·Hz2/3 and 91.53 dB·Hz2/3, respectively, as shown in Fig. 6(b). Hence, the proposed modulation scheme provides around 9.7 dB SFDR improvement relative to the MZM. Compared to simulation, the result corresponds to $\kappa L$ of 4. The grating coupling coefficient $\kappa $ is affected by fabrication imperfections, such as sidewall angle, surface roughness loss, smooth grating edges, and etching depth tolerance. Higher SFDR improvement could be obtained using longer gratings at the expense of higher scattering loss and a slight decrease in RF bandwidth of spiral BGRs. In addition, a phase shifter could be added to the unmodulated arm using active thermo-optic tuning, which is more efficient and stable than EO tuning at DC in Z-cut LN, to mitigate the possible shift between the Bragg and Michelson quadrature wavelengths [48]. The measured noise floor in 1 Hz bandwidth is -154 dBm/Hz, limited by spectrum analyzer and photoreceiver noise. The SFDR could be significantly enhanced by reducing the coupling loss of the device using edge coupling, as well as reducing the noise floor in the measurement link.
Fig. 6. (a) Measurement setup to characterize RF response of the linear modulator. The two-tone test is applied to a single arm for SFDR measurement (blue path), then a differential signal was applied to both mirrors for push-pull operation (green path). EDFA, erbium-doped fiber amplifier; FPC, fiber polarization controller; DUT, device under test. (b) Experimental results of IMD3 SFDR of the MZM and the proposed GAMIM for single arm modulation at 1 GHz. (c) Measured FH of differentially modulated output compared to single arm, validating push-pull arrangement for the device. (d) Percentage modulation depth as a function of applied RF voltage, visualizing the effect of ${V_\pi }$ of the modulator.
The push-pull arrangement of the proposed configuration was validated for linear amplitude modulation applications. Both mirrors were modulated by applying a differential signal using GSGSG RF probe, as shown in the setup in Fig. 6(a). The intensity modulated output was measured using direct detection at the photoreceiver, where the FH RF power is compared to single arm modulation in Fig. 6(c). To characterize the linearity of the output electric field of such a configuration, a coherent demodulation scheme is required, which is out of the measurement scope of this paper. The device could be integrated in the future within coherent photonic links, where high linearity is essentially required.
Based on the measured RF power, the modulation depth of the output analog signal, defined as the percentage ratio between the modulated optical power and the carrier power at the bias point, is plotted as a function of applied RF voltage in Fig. 6(d). The modulated optical power is calculated from the FH RF power using the photoreceiver sensitivity [24]. In order to decrease the required voltage swing of the modulator for a certain output power, ${V_\pi }$ can be reduced in future designs in two ways. First, by reducing the optical bandwidth of the grating reflectivity using lower $\kappa $ gratings, requiring longer reflectors for linear operation $({\kappa L \to \infty } )$. Using the spiral configuration, long grating reflectors could be implemented without significant increase in the device area. Second, by enhancing the tuning sensitivity by decreasing the top SU-8 thickness at the expense of optical loss and device capacitance.
Finally, the frequency response of the linear modulator was measured from 100 MHz to 20 GHz, using the setup shown in Fig. 7(a). Two-port vector network analyzer (VNA) (Agilent N5230A) was employed, where Port-1 was used to generate the modulation signal, and Port-2 was connected to the photoreceiver output to measure small signal EO bandwidth (S21). The input light wavelength was tuned to the quadrature point at $- $3 dB optical transmission, using the tunable laser source. Output results are plotted in Fig. 7(b), demonstrating wide bandwidth up to 18 GHz. The bandwidth of the spiral Z-cut modulator is limited by the R-C time constant, resulting from the structure’s high capacitance, and phase mismatch between optical and RF fields, resulting from different phase shifts acquired by the optical field while traveling through the waveguide [7]. The bandwidth limitation can be addressed either by reducing the device capacitance between top and bottom electrodes using thicker top cladding layer, at the expense of reduced sensitivity, or by changing the RF resistance from 50 Ω using a custom design for the driving stage [36]. The measurement results of the linear modulator are summarized and compared to the reported SFDR on TFLN substrate in Table 1.
Fig. 7. (a) Measurement setup to characterize the frequency response of the GAMIM. VNA, vector network analyzer. (b) Measured small signal response showing EO bandwidth (S21) of the proposed linear modulator.
Table 1. Comparison of the reported modulators on TFLN platform
4. Conclusion
A highly linear, compact and wideband Michelson interferometer modulator assisted by spiral Bragg grating reflectors has been demonstrated on Z-cut lithium niobate substrate. Linearized performance is achieved by modulating the reflectivity of one or both of the long BGRs, compensating for the Michelson interferometer's sub-linear transfer function. The required long gratings are achieved in an ultra-compact form based on spiral-shaped waveguides with width corrugations, allowing for a 3 mm long grating to fit in an 80 µm × 80 µm area. Moreover, the device is modulated using top and bottom electrodes, where SU-8 polymer is used to achieve a perfectly uniform top cladding layer. The proposed modulator achieves an SFDR of 101.2 dB·Hz2/3 for single arm modulation, with 9.7 dB improvement compared to reference MZM on the same substrate. A further reduction in coupling loss and increasing grating length would increase the measured SFDR. In addition, differential modulation is demonstrated for the device, allowing for linear amplitude modulation that could be employed in analog coherent links. The modulation depth of the analog-modulated output was characterized to analyze the required input RF voltage of the linear modulator. The high-speed measurement demonstrates a wide modulation bandwidth of 18 GHz for the compact device, limited by R-C time constant and optical to RF phase mismatch.
Appendix A: Derivation of GAMIM output using single-arm modulation
The electric field at the output of unbalanced GAMIM is given by:
For very long gratings $({\kappa L \to \infty } )$, the $\textrm{tanh}$ function approaches unity. Substituting for the output electric field for single arm modulation (modulating ${r_2}$ only):
After mathematical manipulations:
Such a result implies that optimum linear operation could be reached at the quadrature phase point where $\Delta \varphi = {90^o}$. Hence the output power becomes:
Such perfect linear characteristics would be affected if the device is operated at another shifted frequency from ${f_B}$. The frequency detuning would be, $\delta f(V )= \; \delta {f^{unmod}} + \Delta f.V/2{V_\pi }$, where $\delta {f^{unmod}} = f - f_B^{unmod}$. The electric field at $\Delta \varphi = {90^o}$ will be:
After squaring, the output power is found to be:
Performing a Taylor series expansion in $V/{V_\pi }$ results in:
Thus, we observe the first nonlinear term is quadratic in voltage. Performing a Taylor series expansion in detuning as well shows that this quadratic voltage term scales linearly with the frequency detuning. In contrast, the cubic term in voltage, that mainly affects the IMD3 SFDR, is proportional to the square of detuning.
Finally, we consider the effect of different transmission coefficients in both Michelson arms that may result from fabrication imperfections impact on MMI splitting ratio or bending loss. Assuming ${t_1}$ and ${t_2}$ coefficients in both arms, output electric field becomes:
After squaring, the output power at $\Delta \varphi = {90^o}$ becomes:
Hence, the modulated power is scaled without affecting the linear performance.
Appendix B: Derivation of GAMIM output using differential modulation
Following the same procedure for single arm modulation, the output field for $\kappa L \to \infty $, while driving both mirrors with differential signals, is calculated as:
Considering intensity modulation first, the output power can be calculated by phasor representation as:
After mathematical manipulations:
The best linear operation could be obtained at $\Delta \varphi = {90^o}$, where the power becomes:
As aforementioned, the obtained linearity for intensity modulation has little improvement compared to the MZM case due to the existing nonlinear terms. Considering a scheme where the goal is instead the modulation of electric field amplitude, a purely linear response can be obtained by operating at $\Delta \varphi = {180^o}$. The electric field becomes:
Assuming nonzero $\delta {f^{unmod}}$, the output field at $\Delta \varphi = {180^o}$ will be:
This time, performing a Taylor series expansion in $V/{V_\pi }$ results in:
Regarding the effect of different transmission in Michelson arms; ${t_1} \ne {t_2}$, the calculated output electric field in Eq. (25) becomes:
Thus, the linearity of output electric field is affected, where the nonlinear term is proportional to the difference between both transmission coefficients $({{t_1} - {t_2}} )$.
Funding
National Aeronautics and Space Administration (80NSSC17K052).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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