Real-time free spectral range measurement based on optical single-sideband technique

Xiao Wang; Lishuang Feng; Zhen Zhou; Hui Li; Danni Liu; Qiwei Wang; Liying Liu; Yongze Jia; Hongchen Jiao; Ning Liu

doi:10.1364/OE.26.007494

1. Introduction

Optical resonators have been widely studied devices in many areas of photonics [1], and in many applications such as narrow-linewidth laser sources [2], microcombs [3], optical isolators [4], filters [5], resonant vertical directional couplers [6], and high-performance compact optical gyroscopes [7–9]. In addition to the Q factor and mode volume, precise knowledge of the free spectral range (FSR) is required for applications in which the exact resonant frequencies are important. In particular, for narrow-linewidth laser sources, the high Q optical resonator always acts as a frequency discriminator to provide reference for the laser frequency [10], and the instability of FSR determines the shift of the laser center frequency, causing extra frequency walk-off. As another example, optical resonant gyroscopes require knowledge of the cavity FSR to correctly estimate the Sagnac frequency shift, and understanding the FSR instability would be useful for analyzing the noise distribution and bias stability of the gyro [11]. In addition to estimating the frequency instability, measurement of the FSR as a function of wavelength can provide a convenient method for characterizing resonator dispersion.

Most FSR measurement techniques make use of the intensity response of the resonator. One common method is based on a tunable laser locking system that incorporates a high-accuracy optical wavelength meter. Additionally, a technique based on a PDH error signal has been demonstrated with subhertz accuracy, which is achieved by incorporating a 1-kHz linewidth laser swept in frequency with an acousto-optic modulator [12]. Within the last decade, an optical frequency comb has provided several methods for measuring FSR using knowledge of the comb repetition frequency [13–16]. Moreover, a fast, broadband, and accurate scheme has been achieved by combining an optical frequency comb with a tunable external cavity diode laser, in which the stabilization of the frequency comb sources was used to determine the precision of the measurement [17]. In another method, a single modulator was used in combination with a reference interferometer to measure both the spectral and transverse mode dependence of dispersion in a resonator [18]. While this method of combining a modulator and a reference interferometer is a convenient method for measuring the FSR, the method requires a separate fiber-calibrated Mach–Zehnder interferometer (MZI) and requires laser frequency sweeping whose nonlinearity could introduce systematic measurement error. These techniques are further limited by the laser linewidth and the full-width at half-maximum (FWHM) of the resonator.

In this study, we demonstrate a novel scheme for real-time FSR tracking that is realized by a fiber-based, hybrid, unbalanced Mach–Zehnder modulator (MZM) with an optical single-sideband (OSSB) technique, in order to meet the requirements of real-time FSR measurement in non-traditional application fields (e.g., passive resonator gyroscope [19] and high-performance microwave signal generation [20]). The OSSB technique, which was first demonstrated by Izutsu et al. [21] in the early 1980s, can precisely shift the laser frequency by using the microwave signal that drives the child MZI arms (see Fig. 1) [22, 23]. In the present work, two Pound–Drever–Hall (PDH) loops were implemented to attain the locking of two wavelengths to adjacent cavity modes, in which one used the original laser frequency and another utilized an OSSB frequency-shifted version of the laser. The frequency modulation to amplitude modulation (FM-to-AM) conversion of the resonator introduces an ambiguity that makes it difficult to distinguish the cavity modes of each wavelength based only on the intensity of the drop-light or by using one tone injection to modulate the light. To solve this problem, two different tones are injected into the original and frequency-shifted versions of the laser.

Fig. 1 Schematic diagram of the real-time FSR-tracking system. Top: structure of the hybrid unbalanced MZM; Bottom: experimental system setup. (TNLF laser, tunable narrow-linewidth fiber laser; MZM, Mach–Zehnder modulator; HC, hybrid coupler; PD, photodetector; PM, phase modulator; WRR, chip-based waveguide ring resonator; C₁, C₂, evanescent wave couplers; Demod, demodulation module; RTL, resonance-tracking loop; err. @ Ω_{1, 2}, PDH error signals at Ω_{1, 2}; ADC, analog-to-digital converter; DAC, digital-to-analog converter; DPLO, digital-controlled phase-locked oscillator; FPGA, field-programmable gate array.)

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Systematic and random phase delays in the MZI arms introduce another challenge in tracking the FSR using the OSSB technique as the phase difference of the carrier lights (i.e., the original frequency lights in the master and slave arms in Fig. 1) would cause the interference noise at the output of the hybrid, unbalanced MZM. Such errors, which are proportional to the zeroth-order Bessel function, vary the MZI balance and introduce uncertainties in the PDH signals. To overcome this problem, a frequency dither is injected into the master arm. The phase amplitudes of the two modulation tones and the frequency dither are adjusted such that interference terms are driven near zero [24–26]. When the two cavity modes are steadily locked, the frequency of the microwave signal is equal to the real-time FSR. To demonstrate our method for determining the FSR in real-time, we measured the FSR coefficient of thermal expansion (CTE) of a ring resonator with a Si₃N₄ core and SiO₂ cladding. The fitted expansion result agrees with the simulated value using the two-dimensional finite-element method (2D FEM) solver described in [27]. The resulting technique could be applied in advanced optical inertial setups and microwave photonic signal processing systems.

2. Measurement Process

The schematic diagram of the proposed real-time FSR-tracking setup with a waveguide ring resonator (WRR) is shown in Fig. 1. A tunable narrow-linewidth fiber laser was used as an optical carrier signal. The fiber-based, hybrid, unbalanced MZM consisted of an OSSB modulator with two parallel child MZIs located on the parent’s arms and two phase modulators (PMs) located on the grandparent’s arms. For generating OSSB frequency-shifted light, a direct current (DC) controller was used to adjust the relative optical phases between the child and parent MZI arms, and an electrical 90° hybrid coupler (HC), also known as a quadrature hybrid, allowed the injected microwave signal to be split into two branches with identical power and a 90° phase difference. A chip-based WRR with fiber-coupled connectors served as the optical device-under-test.

Light from the tunable laser was split into two waves at point ①, where one wave was directly injected into PM1 and the other wave was modulated as the OSSB frequency-shifted signal via the OSSB modulator. Each arm of the OSSB modulator had a MZI with identical optical lengths. For generating the OSSB frequency-shifted signal, the MZIs were driven by cosine and sine microwave signals, which were separated by the microwave signal injected into the HC. The relative optical phases between the MZIs (i.e., $φ_{C h 1}$ , $φ_{C h 2}$ , and $φ_{P}$ in Fig. 1) were set as π, π and π/2, respectively, by adjusting the bias voltage through DC control. Each light wave at point ② was modulated by different single-frequency signals (i.e., $Ω_{1}$ and $Ω_{2}$ ) before being introduced into the WRR in order to separate the intensity of the slave wave from the master wave. Here, one frequency dither $Ω_{F D}$ was injected into the master arm to reduce the interference noise.

After the signals dropped out from the cavity, the optical signals detected by the photodetector (PD) were demodulated to be PDH error signals. Not all captured signals in the detector could normally be considered independent processes [28]. In other words, the sensing signal of the master beam would experience interference from the slave beam, and vice versa. Therefore, interference between master and slave should be considered. Moreover, the laser frequency was centered at the resonance of the resonator using the well-known PDH technique in order to track the resonance mode. The OSSB frequency-shifted light, which is called the slave light, was locked on the neighboring resonance mode to the central resonance mode on which the master light was locked. Since the master and slave lights originated from the same laser source, they had the same initial phase, and the FSR could be calculated through the frequency of the microwave signals in real time after accounting for all paths.

Light with frequency $f = ω / 2 π$ can be represented mathematically by its electric field, $E_{0} e^{i ω t}$ , where $E_{0} = 1$ for convenience in the following analysis. Consequently, the phase-modulated electric fields of the master and slave lights at point ③ in Fig. 1 can be expanded using the Jacobi–Anger expansion [29] as

E_{m a s t e r} = e^{i ω t} \cdot α_{m a s t e r} \cdot (1 - κ) \cdot [\sum_{m = - \infty}^{\infty} \sum_{n = - \infty}^{\infty} i^{n + m} J_{n} (β_{P M}) J_{m} (β_{P M}) e^{i (n \cdot Ω_{1} + m \cdot Ω_{F D}) t}]

E_{s l a v e} = e^{i ω t} \cdot e^{i ϕ_{G P}} \cdot α_{s l a v e} \cdot κ \cdot \sum_{m = - \infty}^{\infty} \sum_{n = - \infty}^{\infty} [\begin{array}{l} i^{m} J_{n} (β_{O S S B}) J_{m} (β_{P M}) e^{i (n \cdot ω_{O S S B} + m \cdot Ω_{2}) t} \cdot \\ (e^{- i π} + {(- 1)}^{n}) (1 + i^{n} \cdot e^{- i \frac{π}{2}}) \end{array}]

where

J_{n}

,

κ

,

α_{m a s t e r}

,

α_{s l a v e}

,

ϕ_{G P}

,

β_{O S S B}

,

β_{P M}

,

ω_{O S S B}

,

Ω_{1}

,

Ω_{2}

and

Ω_{F D}

denote the n-th Bessel function of the first kind, the splitting ratio, the overall losses

α

of the master and slave arms, the final phase difference between the master and slave arms, the modulation depths

β

of the OSSB and PM, the angular frequency of microwave signal, the angular frequencies

Ω

of the modulating signals in PM1 and PM2, and the frequency dither, respectively. The phase-modulated field

E_{s l a v e}

had spectral components with frequency shifts of

ω_{O S S B}

,

3 ω_{O S S B}

,

5 ω_{O S S B}

, etc. A series of sidebands were located at one side of the optical carrier due to the value of

β_{O S S B}

.

Figure 2 shows the simplified spectra of the optical signals in Eqs. (1) and (2). By considering the different losses of each arm, we could balance the energy of the master and slave lights by changing the splitting ratio $κ$ . The different frequencies of the PDH error signals at $Ω_{1}$ and $Ω_{2}$ reflected the phase samples of mode1 and mode2, respectively. Therefore, this feature was used to distinguish and lock different modes of the cavity. The key to achieving the system proposed in Fig. 1 is the stability of OSSB frequency-shifted light, which requires precise control and stabilization of the relative optical phases $φ_{C h 1}$ , $φ_{C h 2}$ , and $φ_{P}$ . The versatile bias control technique presented in [30] was used to ensure this stability. Here, low frequency dither tones were applied to the DC bias, and the ratio of the second-order harmonic signal to the fundamental first-order signal from the OSSB modulator output port at point ② in Fig. 1 was used to stabilize the relative optical phases.

Fig. 2 (a) WRR resonance modes for propagating lasers. Three resonance modes are separated by 1 FSR; (b) and (c) show the simplified spectra of the master and slave lights at point ③, respectively.

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The other major parameter for generating the OSSB light was the OSSB modulation depth $β_{O S S B}$ , whose value determined the distribution of OSSB energy on the spectrum. Figure 3(a) shows the variations of $J_{0} (β)$ , $J_{1} (β)$ , $J_{3} (β)$ , and $J_{5} (β)$ with respect to the modulation index $β$ , and the intensity of the slave light at point ② is shown in Fig. 3(b). Ordinarily, the modulation depth is less than 2π due to power limitations of the microwave signals. With the optimal $β_{O S S B} = 1.84$ , we obtained the maximum value of $J_{1} (β_{O S S B})$ . Moreover, the first-order optical spectral component, $ω + ω_{O S S B}$ in Fig. 2(c), reached its maximum at point ②.

Fig. 3 (a) Plot of the Bessel function of the first kind, $J_{n} (β)$ , for integer orders n = 0, 1, 3, 5; (b) The intensity of the OSSB electric fields at point ② with the relative optical phases $φ_{C h}$ and $φ_{P}$ set as π and π/2, respectively.

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Before realizing a practicable system, the interference noise formed by the original master and slave lights (i.e., the carrier lights at $ω$ in Fig. 2) should be reduced. The phase difference between the master and slave arms, $ϕ_{G P}$ , was the major influencing factor of the interference noise, as deduced from Eqs. (1) and (2). The primary method for reducing the interference noise involved reducing the energy of the carrier lights (i.e., carrier suppression) [25]. While the OSSB technique did perform carrier suppression to generate the single sideband, it could be difficult to ensure that the accuracy of the relative optical phases matched the required interference noise suppression due to manufacturing nonuniformity and environmental influences. As a result, further suppression was performed using modulation tones, whose modulation depth $β_{P M}$ was set to the zero point of $J_{0} (β)$ . In particular, $β_{P M}$ = 2.405, as shown in Fig. 3(a). Moreover, the additional frequency dither $Ω_{F D}$ injected into the master arm helped enhance the suppression.

The power of the output light $E_{d r o p}$ was proportional to the square magnitude of the electric field:

\begin{matrix} P_{d r o p} = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} P_{m a s t e r (n, m)} (ω) \cdot e^{i (Ω_{1} t)} + \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} P_{s l a v e (n, m)} (ω, ω_{O S S B}) e^{i (Ω_{2} t)} \\ + P_{b e a t_Ω_{1}} (ω, ω_{O S S B}, ϕ_{G P}) \cdot e^{i (Ω_{1} t)} + P_{b e a t_Ω_{2}} (ω, ω_{O S S B}, ϕ_{G P}) \cdot e^{i (Ω_{2} t)} \\ + (r e m a i n d e r a n d h i g h e r o r d e r t e r m s) \end{matrix}

where,

\begin{matrix} P_{m a s t e r (n, m)} (ω) \propto α_{m a s t e r}^{2} \cdot {(1 - κ)}^{2} \cdot J_{n} (β_{P M}) J_{n + 1} (β_{P M}) J_{m}^{2} (β_{P M}) \\ P_{s l a v e (n, m)} (ω, ω_{O S S B}) \propto α_{s l a v e}^{2} \cdot κ^{2} \cdot J_{n}^{2} (β_{O S S B}) J_{m} (β_{P M}) J_{m + 1} (β_{P M}) \cdot \\ {(- 1 + {(- 1)}^{n})}^{2} (1 + i^{- (n - 1)}) (1 + i^{n - 1}) \\ P_{b e a t_Ω_{1}} (ω, ω_{O S S B}, ϕ_{G P}) \propto α_{s l a v e} \cdot α_{m a s t e r} \cdot κ \cdot (1 - κ) \cdot J_{0} (β_{O S S B}) J_{1} (β_{P M}) \cdot \\ [J_{0}^{2} (β_{P M}) \cdot ε_{O S S B}] \cdot [e^{- i ϕ_{G P}} (1 + i) - e^{i ϕ_{G P}} (1 - i)] \\ P_{b e a t_Ω_{2}} (ω, ω_{O S S B}, ϕ_{G P}) \propto α_{s l a v e} \cdot α_{m a s t e r} \cdot κ \cdot (1 - κ) \cdot J_{0} (β_{O S S B}) J_{1} (β_{P M}) \cdot \\ [J_{0}^{2} (β_{P M}) \cdot ε_{O S S B}] \cdot [- e^{- i ϕ_{G P}} (1 + i) + e^{i ϕ_{G P}} (1 - i)] \end{matrix}

Here,

ε_{O S S B}

is the carrier suppression coefficient of OSSB, which tends to be zero. The terms oscillating at modulation frequencies

Ω_{1}

and

Ω_{2}

, namely,

P_{m a s t e r (n, m)}

and

P_{s l a v e (n, m)}

were the expected PDH error signals. Notably, the undesired power of interference effect,

P_{b e a t_Ω_{1}}

and

P_{b e a t_Ω_{2}}

, which could also be extracted from

P_{d r o p}

when demodulating

P_{m a s t e r (n, m)}

and

P_{s l a v e (n, m)}

, became the frequency tracking error in the PDH loops. However, this effect was considerably minimized by the factor

J_{0}^{2} (β_{P M}) \cdot ε_{O S S B}

. Therefore, the PDH error signals were primarily driven by the light angular frequency, indicating the mismatch between the master and slave lights and the resonant modes of the cavity.

In Fig. 4, the quadrature and in-phase demodulated PDH error signals of the slave light are plotted. The PDH error signals, whose slopes have opposing signs at $Δ ω_{O S S B} = 0$ , were dependent on the frequency gap $Δ ω_{O S S B}$ and the modulation depth $β_{O S S B}$ . At the optimal $β_{O S S B}$ value, the slave PDH signals reach their maximum amplitudes. Comparing Figs. 3(b) and 4, instability regions appeared at higher modulation depths of $β_{O S S B}$ due to the higher order spectral components being generated. These new frequencies inherited different phase sampling, thus interfering with the demodulated PDH error signal [31]. To ensure efficient FSR tracking, $β_{O S S B}$ should not be too high, and the modulations and demodulations should be synchronized in phase. Moreover, the slope of the PDH error signal increased linearly with the resonator finesse (for a given FSR) as a result of a reduction in the resonator FWHM. To obtain similar energies in the master and slave lights, we adjusted the splitting ratio $κ$ of the hybrid unbalanced MZM according to the actual overall loss difference between the two arms, i.e. $κ = 0.1$ with 9-dB loss difference. A real-time FSR measurement system with the OSSB technique could be achieved with $β_{O S S B} = 1.84$ and $β_{P M} = 2.405$ . Therefore, with stabilized locking of the two resonance-tracking loops, the value of $f_{O S S B} = ω_{O S S B} / 2 π$ reflected the real-time FSR.

Fig. 4 Simulations of the demodulated PDH error signals at $Ω_{2}$ from PD. (a) Quadrature, (b)in-phase, where $Δ ω_{O S S B}$ refers to the difference between the OSSB light frequency and the intrinsic resonance mode of WRR.

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3. Experiments and discussion

A prototype was constructed based on Fig. 1. A tunable narrow-linewidth fiber laser with a linewidth of less than 1 kHz was employed as the optical carrier with central wavelength and power of 1550 nm and 45 mW, respectively. This laser was a turn-key single frequency DFB fiber laser system with active wavelength control and fast piezo-electric tuning capability. The laser wavelength can be modulated externally at kilohertz (kHz) modulation bandwidth. A polarization-maintaining isolator was placed between the hybrid, unbalanced MZM and the laser source to eliminate echoed light. After passing through the splitter, the slave light was then injected into the OSSB modulator (FTM7962EP), which had an optical bandwidth of 22.5 GHz and maximum $V_{π}$ at the radio-frequency (RF) port of 3.5 V and DC ports of 14 V. By defining the bias voltages as U_Ch1, U_Ch2, and U_P, we set the DC bias voltages of the OSSB modulator as U_Ch1 = 9.01 V, U_Ch2 = 8.93 V, and U_P = 4.67 V to suppress the optical carrier and unneeded sideband. These bias voltages were controlled using the DC controller mentioned earlier by monitoring the optical power variation to avoid bias drift. The sinusoidal microwave signal emitted from the oscillator and driven to the OSSB modulator was pre-amplified by an electrical amplifier that had a 3-dB bandwidth from 75 kHz to 10 GHz, a gain of 26 dB, and a maximum output power of 23 dBm. The amplified microwave signal was divided into two paths using a broadband electrical 90° HC with an operational bandwidth of 2–12.4 GHz. The two obtained microwave signals were passed into the two RF ports of the OSSB modulator. Then, both lights at point ② were phase modulated before being injected into the WRR, and the half-wave voltages of the PMs were 3.36 V and 3.40 V. Moreover, the data acquisition, signal processing, and automatic control of the laser PDH RTL and the OSSB PDH RTL were realized with FPGAs.

The WRR under test was based on the high-aspect-ratio Si₃N₄ platform (width:thickness > 10:1), which caused a high polarization extinction ratio due to the huge difference in bend loss between the transverse-electric (TE) and transverse-magnetic (TM) modes [32–35]. This in turn minimally affected the polarization transfer function shape under temperature fluctuations (i.e., the peak shape of the resonance, which determined the zero point of the PDH error signal). Further, the interference dip of the TE and TM peaks presented in [36] did not exist in this case, which was beneficial for determining the temperature coefficient using the proposed real-time FSR measurement method. In this study, a high-temperature-annealed Si₃N₄ WRR with a bend radius of 35 mm and core thickness of 40 nm was used to stay in the single mode regime with a waveguide width of 4 μm, as shown in Fig. 5(a). Figure 5(b) shows an illustration of the cross-section of the Si₃N₄ waveguide used to realize the resonator. The bottom cladding was thermally grown SiO₂, and the top cladding was completed with low-pressure chemical vapor deposition (LPCVD) and plasma-enhanced chemical vapor deposition (PECVD) SiO_x.

Fig. 5 (a) Fiber-coupled Si₃N₄ WRR (the red dashed line outlines the ring resonator); (b) Illustration of the cross-section of the Si₃N₄ waveguide.

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To confirm that the $Ω_{1}$ , $Ω_{2}$ , and $Ω_{F D}$ modulation frequencies were optimal in relation to the FWHM, an OSSB frequency sweeping was realized with another independent OSSB modulator to estimate the FWHM of the Si₃N₄ WRR. The OSSB light frequency was swept as the frequency of the microwave signal was linearly changed with a certain laser frequency. As shown in Fig. 6, the sweeping step of the microwave signal was 5 MHz. Note that impulse noises appeared at the edge of each step when the frequency changed, which was caused by PLL hopping before the microwave frequency became accurate and stable. The FWHM value, experimentally measured to be 59.1 ± 0.1 MHz, was the average of many measurements with different sweeping steps. Therefore, the frequencies of the phase modulation tones, $Ω_{1}$ and $Ω_{2}$ , and the frequency dither $Ω_{F D}$ , were chosen as 17.85MHz, 20.28MHz and 31.3kHz, respectively, as determined by the “maximin” theory [37] to avoid spectrum aliasing.

Fig. 6 Resonance signal plot of the Si₃N₄ WRR. The data is obtained by OSSB sweeping with a step frequency of 5MHz.

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The FSR-tracking process was divided into four steps in order to show system improvement with the introduction of different frequency tones. Figure 7 shows the frequency spectra of the signals detected by PD1. At first, only $Ω_{1}$ was in the system, and the master light was locked at mode1 of the WRR. As shown in Fig. 7(a), the power of the PDH error signal at $Ω_{1}$ , which reflected the locking error of PDH RTL, was barely stabilized at a low level as a result of huge interference effects from the optical carriers. Some the interference power existed at the sides of $Ω_{1}$ , which were 8 dB above the noise floor. The subsequent introduction of $Ω_{2}$ led to a good condition in which the signal power at $Ω_{1}$ could be stabilized at −40 dBm, as shown in Fig. 7(b). Meanwhile, higher order terms appeared, which shared the interference effects. After the frequency dither $Ω_{F D}$ was adopted, the spectral lines were broadened, and the light power was further subdivided. As shown in Fig. 7(c), the power of the interference could be shared by higher order terms that were introduced by $Ω_{F D}$ to reduce its effect on the PDH error signals. The mode2 locking process (i.e., OSSB PDH RTL) reduced the signal power at $Ω_{2}$ . The spectra of the stable tracking system is shown in Fig. 7(d), after the slave light was locked at mode2 of the WRR. The power at each frequency in the band was below −40 dBm, and two power regions were formed at the sides of 10 MHz.

Fig. 7 The frequency spectra of signals detected by PD1 when the master light was locked at mode1 of the WRR (RBW = 3 kHz). A Rohde & Schwarz signal and spectrum analyzer was used in the measurement. (a) to (d) show the different states of the tracking process. (a) Only Ω₁ used in the system. Unwanted sidebands appeared at the side of Ω₁. (b) Ω₁ and Ω₂ with their harmonic components. (c) Ω_FD introduced to the system after (b). (d) The master and slave light locked at their respective resonance modes.

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To properly characterize the system performance, we utilized the temperature performance of the Si₃N₄ WRR via the temperature control and measurement setup presented in [36]. First, we estimated and measured the value of the FSR at room temperature. After performing characterization measurements on the sample using a scanning electron microscope (SEM), we assumed three unknown device parameters in the simulation model: refractive index of Si₃N₄ at $λ_{0} = 1550 n m$ ; refractive index of the top cladding layer; chromatic dispersion of Si₃N₄. The index of the bottom thermal SiO₂ cladding was assumed to be 1.444. In the wavelength band of our tunable laser and OSSB signal (1550 ± 0.02 nm), the refractive index of the Si₃N₄ was assumed to have a linear dispersion relation as $n_{c o r e} (λ) = n_{0} + α (λ - λ_{0}) / λ_{0}$ , and the top and bottom cladding indices were assumed to be constants. Table 1 lists the values of the parameters used in the simulation, and the theoretical FSR of the sample was calculated to be about 1.864 GHz at 26.8 °C in the cross-sectional r-z plane of the cylindrical coordinate system [27].

Table 1. Parameters of Materials and the WRR Sample

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Two resonance modes of the cavity sample were locked by the RTLs in order to realize the FSR measurement at room temperature. The microwave frequencies were collected with 100 Hz sampling frequency by using data recording software in the upper computer. Here, the FSR of the sample was measured in real-time by tracking the resonance at ~3.7GHz (i.e., the gap between mode1 and mode2 was 2 × FSR). The recorded FSR-tracking results at 26.8 °C are presented in Fig. 8(a). The FSR was experimentally measured to FSR = 1,844,628 ± 6 kHz; thus, a relative precision of 3.25 × 10⁻⁶ was achieved. Therefore, the finesse and the quality factor of the Si₃N₄ WRR were 31.21 ± 0.05 and (3.211 ± 0.005) × 10⁶, respectively.

Fig. 8 (a) Measurement of FSR at 26.8 °C; (b) Data of each temperature points from 31.6 to 78.8 °C.

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Second, 11 temperature spots were chosen as test points for continuous FSR-tracking between 30 to 80 °C. As shown in Fig. 8(b), 6000 samples were collected at each temperature point with 100 Hz sampling frequency; the mean and confidence intervals are plotted in Figs. 9(a) and 9(b), respectively, using error bars. The tracking precision was limited by the FWHM of the sample, which is well known for the case of narrow laser linewidth, and the stability of the FSR relied on fluctuations of the temperature field around the sample. Depending on the performance of the temperature control setup, the data tended to fluctuation slightly because of the relative stabilization of the temperature field at some temperature points, as shown in Fig. 8(b). Thus, there were some differences in the standard deviation of the measured values at different temperature points, as shown in Fig. 9(b). Even so, the relative precision of the measurements was still relatively stable.

Fig. 9 (a) Plot of FSR vs. temperature of the Si₃N₄ WRR with the least squares regression line (green line) and 95% confidence limits (red dashed lines). (b) Standard deviation of the measured FSRs at different temperatures.

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In the temperature band of the experiment, the FSR of the sample was assumed to have a linear dispersion relation as $F S R (T) = F S R_{0} + ρ_{F S R} \cdot (T - T_{0})$ . The stress-optic effect with temperature was significantly smaller than the thermo-optic effect and was thus ignored in our calculations [38]. Moreover, the thermo-optic coefficients of the materials and linear CTE of the sample radius were assumed [39], as shown in Table 1. We further assumed that the thermo-optic coefficients of the top SiO_x and bottom SiO₂ cladding were the same. Thus, the thermally induced changes in FSR could be written as

ρ_{F S R} = \frac{\partial F S R}{\partial T} = \frac{\partial F S R}{\partial n_{c o r e}} \cdot \frac{\partial n_{c o r e}}{\partial T} + \frac{\partial F S R}{\partial n_{c l a d}} \cdot \frac{\partial n_{c l a d}}{\partial T} + \frac{\partial F S R}{\partial R} \cdot R \cdot (\frac{1}{R} \cdot \frac{\partial R}{\partial T})

The values for the sensitivities of FSR to the radius, core, and cladding indices were determined via simulation using the 2D FEM method presented in [27] and were tabulated in Table 2.

Table 2. Sensitivities of FSR

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Finally, from the least-squares fitting results in Fig. 9(a), the fitted equation should be

y = - 16.735 \times 10^{3} x + 1.8451 \times 10^{9}

where y refers to the FSR, and x is the temperature of the ring resonator. The 95% confidence interval for the parameter

ρ_{F S R}

was

- 16.735 \pm 0.002

kHz/°C. This value was in good agreement with the computed thermal expansion value in Table 2. The accuracy of the tracking depended on the stability of the RF signals and the FWHM of WRR. Improvements to the system could be made with more efficient frequency dithering and a higher quality-factor resonator, which would increase the tracking stability and error signal slope, respectively. Therefore, less electrical amplification would be needed, and a lower thermal noise floor could be realized [10].

Another possible method for tuning the frequency of the laser would be the use of an acousto-optic modulator with an accuracy corresponding to the RF frequency [12]. Compared with an acousto-optic modulator, the proposed use of the OSSB technique has an obvious advantage in terms of the tuning range. The frequency shift of an acousto-optic modulator is limited to several hundreds of megahertz, while the tuning range of the OSSB technique can reach several tens of gigahertz. Thus, the proposed OSSB technique should provide flexible and precise light tuning, which is difficult to realize with other existing technologies [22]. This superiority is desirable in several optical information applications, such as the precise tuning of single-photon frequency and high spectral-density optical communication systems [40]. Therefore, our scheme will complement existing FSR measurement systems based on nonlinear optics and would possibly be suitable for use in other resonator-based systems (e.g., resonator frequency combs, optical frequency division, and passive resonant gyroscopes).

4. Conclusion

In conclusion, we developed a scheme for real-time tracking of a ring-resonator FSR using an OSSB technique. This method was used to measure the FSR CTE of the Si₃N₄ WRR. A theoretical model of the system was established based on the electromagnetic theory of light, and the interference features caused by the hybrid unbalanced MZM were carefully analyzed. This real-time FSR measurement method was applicable for Si₃N₄ WRR with a microwave-rate FSR, reaching a relative precision of 3.25 × 10⁻⁶. The finesse and quality factor of the Si₃N₄ WRR measured by the system were 31.21 and 3.211 × 10⁶, respectively. We have used this system to measure the FSR CTE of the Si₃N₄ WRR, which was $- 16.735 \pm 0.002$ kHz/°C and closely matched the theoretical simulation value. Moreover, quality factor increase on the WRR should improve the system performance, and electronics in closer proximity to such devices will allow for higher loop bandwidths and noise suppression. This approach will be useful for understanding the characteristics of high-aspect-ratio Si₃N₄ waveguides and will provide reference data for other optical applications. Furthermore, we expect that this technology would be important for enabling the use of a resonator in many applications requiring real-time FSR measurement.

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Parameter	Value
α (chromatic dispersion)^a	−0.015
Core thickness (nm)	40
Core width (μm)	4
R (radius of WRR) (mm)	17.5
$n_{{Si}_{3} N_{4}}$ at 1550 nm and 30 °C	1.99
$n_{{SiO}_{x}}$ at 1550 nm and 30 °C	1.444
$\partial n_{c o r e} / \partial T$ (RIU/°C)^a	(2.45 ± 0.09) × 10⁻⁵
$\partial n_{c l a d} / \partial T$ (RIU/°C)^a	(0.95 ± 0.10) × 10⁻⁵
_{$R^{- 1} \cdot \partial R / \partial T$} (RIU/°C)^{b, c} (greater than SiO₂ and less than Si)	(0.59–2.56) × 10⁻⁶

Parameter	Value
Simulated $\partial F S R / \partial n_{c o r e}$ (Hz/RIU)	−1.339 × 10⁸
Simulated $\partial F S R / \partial n_{c l a d}$ (Hz/RIU)	−1.106 × 10⁹
Simulated $\partial F S R / \partial R$ (Hz/m)	−1.022 × 10¹¹
Computed $\partial F S R / \partial T$ (Hz/°C)	(−13.62 ~-19.59) × 10³

Parameter	Value
α (chromatic dispersion)^a	−0.015
Core thickness (nm)	40
Core width (μm)	4
R (radius of WRR) (mm)	17.5
$n_{{Si}_{3} N_{4}}$ at 1550 nm and 30 °C	1.99
$n_{{SiO}_{x}}$ at 1550 nm and 30 °C	1.444
$\partial n_{c o r e} / \partial T$ (RIU/°C)^a	(2.45 ± 0.09) × 10⁻⁵
$\partial n_{c l a d} / \partial T$ (RIU/°C)^a	(0.95 ± 0.10) × 10⁻⁵
_{$R^{- 1} \cdot \partial R / \partial T$} (RIU/°C)^{b, c} (greater than SiO₂ and less than Si)	(0.59–2.56) × 10⁻⁶

Parameter	Value
Simulated $\partial F S R / \partial n_{c o r e}$ (Hz/RIU)	−1.339 × 10⁸
Simulated $\partial F S R / \partial n_{c l a d}$ (Hz/RIU)	−1.106 × 10⁹
Simulated $\partial F S R / \partial R$ (Hz/m)	−1.022 × 10¹¹
Computed $\partial F S R / \partial T$ (Hz/°C)	(−13.62 ~-19.59) × 10³

Real-time free spectral range measurement based on optical single-sideband technique

Abstract

1. Introduction

2. Measurement Process

3. Experiments and discussion

4. Conclusion

References and links

Cited By

Figures (9)

Tables (2)

Equations (6)

Optics Express