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Abstract
We propose and demonstrate an integrated wavemeter capable of accurate and broadband measurements without control or knowledge of the temperature. In our design, interferometers composed of silicon and silicon nitride waveguides enable accurate measurements of an input optical wavelength despite large and rapid temperature fluctuations of 20°C by leveraging the disparity in thermo-optic properties of the waveguides. We derive formulas which resolve the wavelength and temperature ambiguity of the interferometers. The fabricated wavemeter chip is found to have a mean accuracy of 11 pm over an 80 nm range near 1550 nm. To our knowledge, this is the first demonstration of an athermal silicon wavemeter and the lowest measurement error across such a broad wavelength range using silicon photonics. This result may reduce the cost and size of wavemeters used in combination with integrated lasers for optical communications, sensing, and other applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Determining the precise wavelength of a laser is important for applications such as optical communications, spectroscopy, and metrology [1–3]. Wavelength meters (wavemeters) provide such measurements, enabling calibration and stabilization of lasers for these and other applications. Wavemeter implementations have generally relied on either interferometers, optical beating, or wavelength-dependent structures [4]. Approaches using interferometers typically send the input laser light together with light from a reference laser into an interferometer (e.g. Michelson) and measure the output fringes while the optical path length is varied by mechanically moving an element, such as a mirror [5–7]. Another approach uses etalons and an imaging array to measure interference fringes [8,9], and other wavemeters similarly use interferograms or speckle patterns on camera sensors [10–14].
Integrated wavemeters have recently been developed to reduce instrument size and cost and to eliminate moving parts. In one demonstration, an integrated wavemeter based on silicon waveguides used an arrayed waveguide grating (AWG) and photodiode array to achieve a detection error below 20 pm across a 20 nm range [15]. Another report involved modulating the index of silicon waveguides, achieving 73 pm root mean square (RMS) accuracy over 10 nm [16]. In another work, an indium phosphide (InP) wavemeter used two integrated Mach-Zehnder interferometers (MZIs) to attain 20 pm resolution over a 40 nm range [17]. However, these demonstrations face limitations due to temperature variations influencing the index of refraction in the waveguides, as silicon and InP have high thermo-optic coefficients (TOCs) of 1.8×10−4/°C and 2×10−4/°C, respectively. Recent works attempted to address this limitation by choice of waveguide material or confinement. In one paper, an interferometer for an on-chip wavemeter included a 62 cm length silicon nitride (Si3N4, TOC ≈ 2.4×10−5/°C) waveguide in order to produce 2.4 pm period fringes and reduced temperature dependence, but the design does not resolve wavelength ambiguity over a large range [18]. In [19], an integrated silicon wavemeter covered the C-band using multiple MZIs and leverages the low-confinement TM mode to reduce temperature sensitivity, but sub-degree temperature control is still required.
Several studies have proposed methods for achieving athermal devices. Passive athermality has been shown by using a material with a negative TOC to balance the waveguide core’s positive TOC [20,21] or by choosing a low-TOC material [22]. However, such materials are not typically available in silicon photonic foundry processes. Athermal MZIs were demonstrated by precisely controlling the waveguide dimensions, but the temperature-stability is relatively narrowband [23,24]. Ideally, a wavemeter could achieve both athermal and broadband performance with potential for integration alongside lasers on a single chip [25–27].
Here we demonstrate an integrated athermal silicon wavemeter requiring neither control nor knowledge of the temperature [28]. Our wavemeter includes MZIs of varying delay lengths to cover a large wavelength range with high resolution, and by using both silicon and Si3N4 waveguides, we can isolate and subtract thermal effects in order to determine a laser’s wavelength at an unknown temperature. The wavemeter includes the MZIs in photonic circuits with integrated photodiodes on a single silicon chip, enabling wafer-scale manufacturing of low-cost wavemeters. Our calibrated wavemeter has a measured average accuracy of 11 pm across over 80 nm, which remains consistent for temperatures fluctuating from 20°C-41°C. The robustness, accuracy, and range of the wavemeter make it attractive for both optical communications and sensing.
2. Theory and design
2.1 Wavemeter using interferometers with 90-degree hybrids
Our integrated wavemeter uses asymmetric MZIs to measure the input signal’s wavelength. Each MZI splits its input into two arms, with the longer one including a path length difference $\mathrm{\Delta }L$ [ Fig. 1(a)]. The two arms connect to a 90-degree hybrid coupler to measure the phase difference between them [29]. This absolute phase difference $\varphi $ depends on the wavelength $\lambda $ and temperature T and is given by
where ${n_{\textrm{eff}}}({\lambda ,T} )$ is the effective index of the MZI waveguides. To retrieve the wavelength from the phase measurement, a linear approximation of the wavelength and temperature dependence around central values ${\lambda _0}$ and ${T_0}$ can be used:
Fig. 1. MZI phase and wavelength measurement. (a) Diagram of an integrated MZI for phase measurement. (b) Spectrum of a first MZI with a small $\mathrm{\Delta }{L_0}$ and large FSR. The red dots indicate the wavelength approximated by the photodetector current measurements ${X_0}$ and ${Y_0}$. (c) Relative phase inferred from the first MZI’s photocurrents. (d) Spectrum of a second MZI with a large $\mathrm{\Delta }{L_1}$ and small FSR. (e) Relative phase inferred from the second MZI’s photocurrents and from the order m as determined by the phase of the first MZI.
To simplify the notation, we define the following variables:
$a \equiv {n_{\textrm{eff}}}({{\lambda_0},{T_0}} )$, the effective index at ${\lambda _0},{T_0}$,
$b \equiv {\left. {\frac{{\partial {n_{\textrm{eff}}}}}{{\partial \lambda }}} \right|_{{\lambda _0},{T_0}}}$, the first order wavelength dispersion,
${n_g} \equiv a - {\lambda _0}b$, the group index,
$\theta \equiv {\left. {\frac{{\partial {\textrm{n}_{\textrm{eff}}}}}{{\partial \textrm{T}}}} \right|_{{\lambda _0},{T_0}}}$, the TOC,
$\mathrm{\Delta }T \equiv T - {T_0}$, the temperature relative to ${T_0}$, and
$\mathrm{\Delta }\lambda \equiv \lambda - {\lambda _0}$, the wavelength relative to ${\lambda _0}$.
The phase difference at each interferometer output can be expressed as
The order of the MZI can be determined by using a second MZI. We previously demonstrated that the unambiguous wavelength range could be extended by cascading several interferometers [17]. An MZI with a small $\mathrm{\Delta }{L_0}$ [large free-spectral range (FSR)] can be used to identify the order of another MZI with a larger $\mathrm{\Delta }{L_1}$ (smaller FSR), as shown in Fig. 1(b-e). Figure 1(b) shows that the two photocurrent measurements from the first MZI (${X_0}$ and ${Y_0}$) identify a single wavelength and phase estimation [Fig. 1(c)]. The second MZI’s photocurrent measurements (${X_1}$ and ${Y_1}$) coincide at many wavelengths [red dots in Fig. 1(d)], making the wavelength determination ambiguous. However, the first MZI’s phase can determine the order of the second MZI, identifying a single, precise wavelength [Fig. 1(e)]. This allows the wavemeter to benefit from the large range of the former MZI while leveraging the higher accuracy of the second MZI. The temperature of the device must be accurately monitored and controlled, though, in order to maintain the accuracy of the wavelength measurements.
2.2 Athermal wavemeter
We remove the temperature dependence of the wavemeter by using MZIs with waveguides having different TOCs. Figure 2 shows a set of MZIs where the first one has waveguides with TOC θ2 and the second MZI has different waveguides with TOC θ3. By comparing the responses of each MZI subject to an unknown change in temperature, it is possible not only to determine the temperature, but also to account for the induced thermo-optic spectral shift when calculating the wavelength, as explained by the following derivation.
The phase measured at each interferometer can be expressed as in Eq. (4). Because the phase is proportional to the optical frequency $\nu = c/\lambda $, we will now mainly use frequency rather than wavelength in the following calculations. The phase differences ${\varphi _2}$ and ${\varphi _3}$ can then be expressed as the system of two equations
The ultimate accuracy of the proposed wavemeter depends mainly on the TOCs and delay lengths. Assuming the main source of uncertainty comes from the electrical measurements of the detectors’ photocurrents, the uncertainty, i.e., the accuracy, of the optical frequency measurement can be expressed as (see Supplement 1, section B):
2.3 Design of the athermal silicon wavemeter
We implement the proposed athermal wavemeter on an integrated platform using silicon and Si3N4 MZIs. The wavemeter design is shown in Fig. 3(a). Input light from a laser is split using multimode interferometers (MMIs) and directed to four MZIs. One MZI has a small delay ΔL0 corresponding to a large FSR0, setting the maximum wavelength range of the wavemeter. The next MZI has an intermediate delay ΔL1, and the third and fourth MZIs have a long delay ΔL2= ΔL3. The first stage resolves the ambiguity in identifying the relevant order of the intermediate stage, and then the intermediate stage is used to identify the order of the long delay MZIs. The first three MZIs are composed of silicon, while the fourth uses Si3N4 waveguides. The phase differences at the outputs of the third and fourth MZIs can then be used to make temperature-independent wavelength measurements using Eq. (11), where θ2 and θ3 correspond to the TOCs of silicon and Si3N4 respectively. The phases are measured using 90-degree hybrid MMIs followed by integrated germanium photodiodes. Note that while we vary the TOC using different materials, it is also possible to vary the width of waveguides of a single material in order to obtain two TOCs. However, this alternative may not easily achieve a large TOC difference since waveguides that are too narrow suffer greater bending losses, and waveguides that are too wide become multimode.
Fig. 3. (a) Diagram of athermal wavemeter design. (b) Photograph of assembled device. (c) Microscope image of the wavemeter chip.
We design the wavemeter’s MZIs to enable wavelength measurement over a large bandwidth with high accuracy. The delay lengths and FSRs at 1550 nm of the four MZIs are listed in Table 1. To cover a wide wavelength range for communications and sensing applications, the FSR of the first MZI is designed to be 94 nm, which corresponds to a 6 µm delay length difference. The following two MZIs successively decrease in FSR by a factor of ∼8.6, which empirically ensures reliable determination of the MZI orders, accounting for possible phase errors in the first stages. The final two MZIs, composed of silicon and Si3N4, have the finest resolution and are used to measure the wavelength according to Eq. (11). We estimate that an accuracy of 10 pm is achievable using these MZI dimensions, based on Eq. (12) and assuming a phase error of 0.02 rad.
Table 1. Mach-Zehnder interferometer parameters
The integrated wavemeter is fabricated on silicon-on-insulator (SOI) wafers using a standard silicon photonic foundry process. The silicon waveguides are 450 nm×220 nm in size and can couple to the 1 µm×400 nm Si3N4 waveguides using tapered evanescent couplers. These waveguide dimensions are chosen for good confinement of the fundamental transverse electric (TE) mode and low bending loss. The 1×2 mm chip is assembled for testing by attaching an optical fiber to the input and wirebonding the electrical outputs to a fan-out board [Fig. 3(b-c)]. A thermoelectric cooler (TEC) is attached to the substrate underneath the chip along with an accompanying thermistor to assist in characterizing and calibrating the wavemeter’s thermal response.
3. Results
3.1 Wavemeter calibration
To perform a one-time calibration of the wavemeter, we extract linear fits of the phase responses of each MZI across the full wavelength range using an external tunable laser. First, we inject light from the laser, sweeping its wavelength across the measurement range while measuring the photodetector currents. We estimate an insertion loss of 5.5 dB to the detectors, including 2 dB coupling loss from the fiber. Figure 4 shows the current measured at the output of each balanced detector pair as a function of wavelength from 1500 to 1580 nm in 10 pm steps, at a fixed TEC temperature of 25°C. For each MZI, we obtain two sine curves in quadrature with a periodicity (FSR) depending on the optical delay. The distortions visible in the sinusoids likely originate in imperfections of the 90-degree hybrid MMIs and the response imbalance of the photodiodes. However, when we calculate the phase from these quadrature curves, the distortions largely disappear. The four top graphs in Fig. 5(a-d) show the phases unwrapped on the calibration range $\mathrm{\Delta }\varphi $ in radians, which are related to the absolute phases by
where ${m_{\textrm{ref}}}$ is the MZI order at the origin of the calibration range ${\nu _0}$ (corresponding to the shortest wavelength). Even though the x-axes use wavelength for convenience, we apply a linear fit to the measurements as a function of the optical frequency, i.e., $\mathrm{\Delta }{\varphi _i}(\nu )= {A_i} + {B_i}\nu $, where i is the respective MZI. The four bottom graphs [Fig. 5(e-h)] represent the error between the measurement and the linear fit, which is typically lower than 0.2 radians. Note that the linear fit errors of the first two MZIs have no effect on the accuracy of the final wavelength measurement, as long as they do not exceed the FSR of the successive stages (which is ensured through proper ratio of ΔL).Fig. 4. Photodetector current measurements from each pair of balanced photodiodes (X and Y) for (a) MZI-0 (b) MZI-1 (c) MZI-2 and (d) MZI-3.
Fig. 5. (a-d) Measured unwrapped phases $\; \Delta \varphi $ for MZI-0 to MZI-3. (e-h) Measurement deviations from the linear fits of MZI-0 to MZI-3 phases.
While the wavemeter operates athermally during regular operation, we must perform a one-time calibration to extract the temperature parameters. We accomplish this by measuring linear fits at several different temperatures. Each linear fit provides two parameters ${A_i}({\mathrm{\Delta }T} )$ and ${B_i}({\mathrm{\Delta }T} )$, such that
Fig. 6. (a-d) Measurement and linear fit of A parameter versus temperature for MZI-0 to MZI-3. (e-h) Measurement and linear fit of B parameter across temperature.
The ${C_{ij}}$ coefficients can be related to the physical parameters as follows:
3.2 Wavemeter operation
We obtain the optical frequency from the MZI phase measurements using successive calculations. Once the calibration parameters have been determined, the procedure first uses MZI-0, which has the largest FSR, to make a first approximation of the optical frequency and thus determine the interferometer order of MZI-1. The phase measurement of MZI-1 is then used to determine the orders of MZI-2 and MZI-3. With these last two MZI orders known, we finally use the theoretical framework from Section 2 to make a temperature-independent calculation of the optical frequency $\nu $ from the phases of MZI-2 and MZI-3.
This procedure involves calculating relative frequency from the measured phase, as well as calculating the subsequent MZI’s order from this frequency approximation. After first retrieving the relative phase $\mathrm{\delta }{\varphi _i}$ from the photodetector currents [Eq. (5)], the relative frequency $\mathrm{\delta }{\nu _i}$ is then calculated using
Fig. 7. Plot of MZI phase measurement, displaying the frequency offset and interferometer order variables. The frequency offset ${\delta }\nu $ can be visualized as the sum of two offsets ${\delta }\nu {^{\prime}}$ and ${\delta }\nu ^{\prime \prime} $.
We calculate the unknown laser frequency by iteratively applying Eqs. (20)–(23) to each MZI. The first approximation $\mathrm{\Delta }{\nu _0}$ from MZI-0 determines the order of MZI-1. We then compute a second, more accurate approximation of the frequency using $\mathrm{\Delta }{\nu _1}$, which we use to determine the orders of the final two MZIs with
3.3 Wavelength measurement results
We measure the wavelength of a tunable laser across an 80 nm range using the wavemeter operation procedure described above. In the experimental setup, a Photonetics Tunics external cavity tunable laser’s output is coupled to the wavemeter chip using an optical fiber. The laser is tuned in 800 steps between 1500 nm and 1580 nm, while the eight wavemeter output currents are measured using an Agilent switch unit connected to a Keithley 2000 multimeter. The TEC temperature was treated as unknown, so its value was ignored for the computation of the measured wavelength. The temperature set point (22.5°C) was chosen not to correspond to values chosen during the calibration. Applying the procedure from Section 3.2 to calculate the laser wavelength from the MZI phases, we obtain the wavemeter’s measurement of the wavelength compared to the known value from the laser [Fig. 8(a)], which closely matches a line. The difference between the measured and set wavelengths is plotted in Fig. 8(b) and also corresponds to the accuracy, with the highly accurate (<5 pm repeatability) laser serving as the reference. The deviation from the linear model has a maximum amplitude of 200 pm and mean error (accuracy) of 84.6 pm. However, we can apply a correction to increase the accuracy even further.
Fig. 8. (a) Wavelength measurement based on linear model compared to the known input laser wavelength. (b) Deviation of the measurement from the known laser wavelength.
By correcting for reproducible errors in the calibration, the wavemeter accuracy improves beyond that of the initial linear model. Figure 8(b) shows that the linear model’s error has significant periodic components. This error includes detector imbalances and other fabrication defects such as spurious reflections (for example, the period of the oscillations in Fig. 8(b) corresponds to the length of the 90-degree hybrid), which can all be systematically corrected from the measurement because they are predictable. To do so, we store the measurement error of each MZI across the wavelength range at several temperatures. During wavemeter operation, this stored error can be subtracted from calculations made using the linear model, i.e., Eq. (20). Initially, an arbitrary value for the unknown temperature +/- 10°C may be used in the Section 3.2 procedure, producing a reasonably accurate estimation of the wavelength and temperature. The temperature itself can be obtained with the formula
Fig. 9. (a) Wavelength measurement using additional corrections compared to known input laser wavelength. (b) Measurement error (accuracy) with the corrections applied across the wavelength range.
The wavemeter accuracy is consistent across a large temperature range, when subjected to rapidly changing temperatures, and for both high and low input power. We measure the mean error across three temperatures unknown to the wavemeter and covering over a 20°C range (Fig. 10). The RMS accuracy stays at 10–12 pm at both ends of the temperature range tested. We also vary the input optical power to the wavemeter from −11.5 dBm (71 µW) to 18 dBm (63 mW) and again find consistent accuracy of 11.0 pm on average. We next monitor the dynamic response of the wavemeter when subjected to a rapidly shifting temperature. To perform a fast acquisition, the eight photocurrents were measured simultaneously at 8.3 kS/s using a National Instruments PXIe 6238 eight-channel acquisition module. With the input laser set to a fixed wavelength, the TEC under the wavemeter chip is switched from 21°C to 39°C (Fig. 11). During this temperature shift, which takes less than five seconds, the wavemeter continues to measure the wavelength and the temperature. Figure 11(b-c) show the wavelength measurement error has negligible fluctuations even as the temperature drifts at ∼5°C/s. The mean error during the temperature ramp at this wavelength is just 4.1 pm, while after the temperature settles, the error is 1.0 pm. This robustness to environmental perturbations demonstrates the benefit of an athermal integrated wavemeter.
Fig. 11. (a) Temperature calculated by the wavemeter during a large fluctuation. (b) Measured wavelength during the temperature shift with the input laser set at 1550 nm. (c) Wavelength measurement error during the temperature shift.
4. Discussion and summary
We expect that the wavemeter’s capabilities may exceed the current demonstration. The measurement range was limited by the available equipment and should in fact cover the full FSR of MZI-0, which is 94 nm. Additionally, improved assembly with high-speed analog-to-digital convertors (ADCs) and transimpedance amplifiers (TIAs) should enable characterization of the ultimate measurement rate. We expect nanosecond-scale measurements are attainable with moderate input power. The germanium photodiodes are capable of GHz operation and thus should not limit the speed in this regime. Next, the wavemeter’s operational temperature range and measurement accuracy may be optimized for a particular application by properly adjusting the MZI dimensions, i.e., the FSRs. Thanks to the chip-scale integration of waveguides, much longer delay lengths may be attained in a compact area with low loss, leading to improved measurement accuracy, while adjusting the shortest delay lengths allows control of the wavelength range. In Supplement 1, section D, we derive equations explaining that there is currently a trade-off between the operational temperature range (∼20°C in our case) and the accuracy, ultimately finding that only a rough estimate of the temperature is needed to overcome the trade-off and extend the maximum temperature deviation significantly. Finally, with the corrected phase errors’ magnitude now known, a larger ratio between the FSRs would allow for either a larger wavelength range or elimination of the intermediate stage without sacrificing accuracy. Nevertheless, in the current design we have achieved a favorable combination of accuracy, wavelength range, and temperature-independence.
We have proposed and demonstrated the first athermal integrated wavemeter using silicon photonics. By including interferometers composed of two different materials, our design removes the temperature sensitivity present in previous integrated wavemeters. We show that our wavemeter chip determines the wavelength of a laser with an average accuracy of 11 pm over an 80 nm range. This represents the lowest measurement error across such a broad wavelength range using silicon photonics, to the best of our knowledge. Such accuracy is maintained for temperature changes of at least 20°C, including a rapidly fluctuating temperature. In comparison, without the athermal design, the longest silicon MZI would cause a measurement drift of ∼1,200 pm over 20°C. This design enables low-cost production of robust, integrated wavemeters for an array of applications. The broadband and accurate measurements will be useful in future integrated sensors in combination with a scanning laser. Additionally, the wavemeter could help to stabilize tunable lasers used in communication networks.
Acknowledgments
The authors thank Michael Eggleston, Po Dong, Argishti Melikyan, Andrea Blanco-Redondo, Mark Earnshaw, Jean-Guy Provost, Alexandre Shen, and Joan Ramirez for productive conversations.
Disclosures
The authors declare no conflicts of interest.
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.
Supplemental document
See Supplement 1 for supporting content.
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