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Abstract
Optical frequency combs generated from microresonators (especially microresonator soliton combs) have been attracting significant attentions because of the potential to be fully chip-scale. Among various promising applications of soliton combs, coherent optical communications and mm/THz wireless communications require low phase noise of the comb modes and low relative phase noise between the comb modes, respectively. Here, we measure the phase noise of a soliton comb, investigating how the thermorefractive noise of a microresonator influences on the phase noise. We observe the quadratic increase of the phase noise of the comb modes, as the comb mode number, counted from the wavelength of a pump cw laser, increases. In addition, we measure the relative phase noise between the comb modes, showing less influence of the phase noise of pump cw lasers by comparing soliton combs generated from pump cw lasers with low and large phase noise.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent advances in fabricating high-Q microresonators have enabled the minimization of optical frequency combs, to be potentially chip-scale, highlighted as microcombs [1,2]. In particular, dissipative Kerr microresonator soliton combs (hereafter called soliton combs), which is a mode-locked state, produce highly coherent optical frequency combs with ultra-short pulse trains [3]. The comb mode spacing of the soliton combs ranges from tens of GHz to THz, allowing easy access to each comb mode. Large comb mode spacing has been utilized in applications such as coherent optical communication [4], ultra-fast ranging [5,6], THz-wave generation [7], and astrophysical spectrometer calibration [8,9].
Phase noise of soliton combs is very important for coherent optical communications [4], LIDAR [10,11], and mm/THz wireless communications [7,12]. Soliton combs are used as multi-wavelength carriers for coherent optical communications and LIDAR, where the phase noise of the comb modes ($L_\textrm {mode}(f)$, note that $L_*(f)$ indicates the single-sideband (SSB) phase noise power spectrum density (PSD) of * in this paper) of the soliton combs is one of the most important parameters. Phase noise limits the measurable distance for LIDAR and the amount of transmitted information for coherent optical communications. For mm/THz wireless communications, high frequency carriers in mm/THz region are generated from the soliton combs through the opto-electric conversion [7]. The phase noise of the generated carriers should be small to put as much as information on the carriers in the same way as coherent optical communications. Unlike the case of coherent optical communications, the relative phase noise between the comb modes ($L_\textrm {relative}(f)$) is important for mm/THz wireless communications, because the relative phase noise sets the lower limit of the generated carriers.
$L_\textrm {mode}(f)$ and $L_\textrm {relative}(f)$ are influenced by several noises such as the phase and intensity noises of pump cw lasers, the mechanical and thermal noises of microresonators [13]. When the free spectral range (FSR) of microresonators is a few tens of GHz, the phase and intensity noises of pump cw lasers limit $L_\textrm {mode}(f)$ and $L_\textrm {relative}(f)$ [14–16]. On the other hand, when FSR of microresonators is large (e.g. > 100 GHz), thermorefractive noise [17], which is the fluctuation of the refractive index caused by thermal noise, cannot be ignored due to the small mode volume of the microresonators. Indeed, very recently, thermorefractive noise was observed in microresonators based on Si$_3$N$_4$ (SiN), which are one of the most promising platforms for fully integrated microcombs because of CMOS compatibility. Drake et. al. [18] observed the increase of the phase noise of the carrier envelope offset frequency of a soliton comb ($L_\textrm {ceo}(f)$) with 1 THz FSR, compared with a pump cw laser. In addition, Huang et. al. [19] numerically and experimentally showed that the thermorefractive noise influences on the phase noise of a cw laser passing through microresonators with the FSR of 88 GHz, 100 GHz, 200 GHz, and 1 THz. However, the influence of the thermorefractve noise on the comb modes and between the comb modes has not been reported, to the best of our knowledge, despite the importance of them for coherent optical communication, LIDAR and mm/THz wireless communication.
In this paper, we measured the phase noise of the comb modes and relative phase noise between the comb modes of a soliton comb generated from a microresonator with about 540 GHz FSR with investigations of the influence of the phase and amplitude noise of a pump cw laser. We observed the quadratic increase of the phase noise of the comb modes depending on the harmonic number of the soliton comb when the phase noise of the pump cw laser is smaller than the thermorefractive noise of the microresonators. We also showed the relative phase noise between the comb modes of the soliton comb has not been so much influenced by the phase noise of pump cw lasers, indicating mass productive, cost-effective cw lasers (e.g., distributed feedback (DFB) lasers) could be used for mm/THz wireless communications without sacrificing the performance of wireless communications. In addition, we showed the influence of the amplitude noise is negligible, which is different from soliton combs with small mode spacing.
2. Results
2.1 Experimental setup
The experimental setup is shown in Fig. 1(a). Pump cw laser is either an external cavity diode laser (ECDL, linewidth < 10 kHz) or a DFB laser (linewidth $\approx$ 1 MHz) with the oscillation wavelength of around 1548 nm. The pump cw laser passes through a dual-parallel Mach-Zehnder modulator (DP-MZM), which is operated in the carrier-suppressed single-sideband mode, for the rapid scanning of the wavelength of the pump cw laser [20,21]. The output from the DP-MZM is amplified by an Er-doped fiber amplifier (EDFA) with one forward and two backward pump laser diodes (LDs). The maximum output power from the EDFA is approximately 500 mW. The output from the EDFA is coupled into a high-Q microresonator based on SiN through a lensed fiber. The FSR and Q of the microresonator are approximately 540 GHz and 10$^6$, respectively. The output from the microresonator is split into two outputs. One is used to measure the optical spectrum of the generated soliton comb. The other is directed to a bandstop filter to reject the residual of the pump cw laser. The output from the bandstop filter is used to measure the phase noise of the soliton comb modes ($L_\textrm {mode,ECDL/DFB}(f)$) and relative phase noise between the comb modes ($L_\textrm {relative,ECDL/DFB}(f)$). Figure 1(b) shows the experimental schematic used to measure $L_\textrm {mode,ECDL/DFB}(f)$ and $L_\textrm {relative,ECDL/DFB}(f)$, separately. $L_\textrm {mode,ECDL/DFB}(f)$ is measured by a delayed self-heterodyne interferometer, which consists of an imbalanced Mach-Zehnder interferometer (MZI) with an acousto-optic modulator (AOM) in one arm of the MZI. An optical bandpass filter is installed before photo detection to leave only one comb mode. By measuring the SSB phase noise PSD of the photo detected signal ($|H|^2L_\textrm {mode,ECDL/DFB}(f)$, where $H$ is the transfer function of the imbalanced MZI), $L_\textrm {mode,ECDL/DFB}(f)$ is obtained. For the measurement of $L_\textrm {relative,ECDL/DFB}(f)$, a two-wavelength delayed self-heterodyne interferometer (TWDI) [22–24] is used. The imbalanced MZI is exactly the same for the measurement of $L_\textrm {mode,ECDL/DFB}(f)$, but the TWDI uses two outputs from an imbalanced MZI. Each output passes an optical bandpass filter, both of which pass through a single comb mode at different wavelengths. By mixing the photo-detected signals followed by a FFT analyzer, the quantity $|H|^2N^2 \times$ $L_\textrm {relative,ECDL/DFB}(f)$, where $N$ is the separation of the comb mode number between the two outputs from the optical bandpass filters, can be measured, while the carrier envelope offset frequency of the soliton comb is cancelled out. Details of the TWDI are shown in Ref. [23,24]. Figure 1(c) shows the optical spectrum of the soliton comb. The comb spacing corresponds to the FSR of the microresonator with a sech$^2$ envelope, indicating that the soliton comb is a single soliton state.
Fig. 1. (a) Schematic of the experimental setup to generate a soliton comb. VCO: voltage-controlled oscillator, DM-MZM: dual-parallel Mach-Zehnder modulator, EDFA: Er-doped fiber amplifier, OSA: optical spectral analyzer. (b) Schematic of the experimental setup to measure the phase noise of the soliton comb modes and relative phase noise between the comb modes. AOM: acousto-optic modulator, OBPF: optical bandpass filter, ESA: electronic spectrum analyzer, FFT: FFT analyzer. (c) Optical spectrum of a soliton comb.
2.2 Phase noise of the comb modes
Figure 2(a) shows $L_\textrm {mode,ECDL}(f)$ at different comb mode numbers. The harmonic number is counted from the mode at the wavelength of the pump cw laser (1548 nm for Fig. 2(a)). The phase noise of the ECDL is also shown (the black curve in Fig. 2(a)). $L_\textrm {mode,ECDL}(f)$ increases as the harmonic number increases. The excess noise can be the themorefractive noise of the microresonator, as indicated in Ref. [18] and Ref. [19]. To investigate a formula of the thermorefrative noise on the comb modes, a single-tone amplitude noise is added to the ECDL, perturbing the FSR of the microresonator through the conversion from the amplitude to refractive index fluctuation. Because both amplitude and thermorefractive noises fluctuate the FSR of the microresonator through the refractive index change, the added amplitude noise can be an emulator of thermorefractive noise. Experimentally, the amplitude noise is added by modulating the pump current of the LD of the EDFA. Modulation frequency ($f_m$) and depth are 10 kHz and about $\pm$ 0.2 %($\pm$ 0.4 mW for 200 mW on chip optical power), respectively. Figure 2(b) shows the SSB phase noise PSD of the $|H|^2L_\textrm {mode,ECDL}(f_m)$ at the modulation frequency. $|H|^2L_\textrm {mode,ECDL}(f_m)$ is quadratically increased as the harmonic number increases. The blue curve in Fig. 2(b) is an numerical curve of $C +20 \times \textrm {log}|k|$, where C and k are an offset and harmonic number, respectively. We implement the same measurements with different pump wavelength at 1562 nm. Results are shown in Figs. 2(c) and (d). Independent of the wavelength of the ECDL, the same behavior as Figs. 2(a) and (b) is observed. The quadratic increase of the phase noise of the comb modes can be explained as follows. The phase noise of the (n + k) th comb mode in the time domain ($\varphi _{n+k}(t)$) can be expressed as [25]
Here $n$ is the comb mode number at the wavelength of the pump cw laser. $\varphi _\textrm {rep}(t)$ and $\varphi _\textrm {ceo}(t)$ are the phase noise of the repetition frequency and carrier envelope offset frequency of the soliton comb, respectively. Because the phase noise of the nth comb mode should be very close to the phase noise of the pump cw laser, we can simply assume $\varphi _{n}(t) = \varphi _\textrm {cw}(t)$, where $\varphi _\textrm {cw}(t)$ is the phase noise of the pump cw laser. With this assumption, Eq. (1) can be written as For the SSB phase noise PSD, Eq. (2) can be expressed as Here, $L_\textrm {rep}(f)$ should be equal to $L_\textrm {relative}(f)$, in principle. When $L_\textrm {cw}(f)$ is smaller than $L_\textrm {rep}(f)$ as in the case of the ECDL, $L_{n+k}(f)$ is dominated by $k^2L_\textrm {rep}(f)$. This is the reason for the quadratic increase of the phase noise as shown in Figs. 2(b) and (d). As Eq. (3) was confirmed for the amplitude noise, thermorefractive noise should also follow Eq. (3), because both the amplitude noise and thermorefractive noise perturbs the FSR through the fluctuation of the refractive index.Fig. 2. (a) Single sideband (SSB) phase noise power spectrum density (PSD) of the comb modes of the soliton comb generated from the ECDL with the wavelength of 1548 nm at the −1st, (red), −3rd (green), −6th (blue), and −9th (purple) harmonics. SSB phase noise PSD of the ECDL is also shown (black). (b) (Red) SSB phase noise PSD of the $|H|^2L_\textrm {mode,ECDL}(f_m)$ at the modulation frequency ($f_m$) when the pump current of the LD for the EDFA is modulated with a single tone at $f_m$. Pump wavelength is about 1548 nm. Numerical curves of $C +20 \times \textrm {log}|k|$ is also shown (blue). (c) Single sideband (SSB) phase noise power spectrum density (PSD) of the comb modes of the soliton comb generated from the ECDL with the wavelength of 1562 nm at the +1st, (red), +3rd (green), +6th (blue), and +9th (purple) harmonics. SSB phase noise PSD of the ECDL is also shown (black). (d) (Red) SSB phase noise PSD of the $|H|^2L_\textrm {mode,ECDL}(f_m)$ at the modulation frequency when the pump current of the LD for the EDFA is modulated with a single tone at $f_m$. Pump wavelength is about 1562 nm. Numerical curves of $C +20 \times \textrm {log}|k|$ is also shown (blue).
When $L_\textrm {cw}(f)$ is larger than $L_\textrm {rep}(f)$, the quadratic increase of the phase noise of the comb modes should not be observed. To confirm this, a soliton comb is generated from a DFB laser with much larger phase noise than the ECDL. Figure 3 shows $L_\textrm {mode,DFB}(f)$ at the different harmonic numbers. The phase noise of the DFB laser is also shown (the black curve in Fig. 3). Unlike Figs. 2(a) and (c), as expected, $L_\textrm {mode,DFB}(f)$ doesn’t depend on the harmonic numbers.
Fig. 3. SSB phase noise PSD of the comb modes of the soliton comb generated from the DFB laser with the wavelength of 1548 nm at the −1st, (red), −3rd (green), −6th (blue), and −9th (purple) harmonics. SSB phase noise PSD of the DFB laser is also shown (black).
2.3 Relative phase noise between the comb modes
Carriers at mm/THz frequency can be generated by directing a soliton comb into a photo detector through opto-electric conversion. For mm/THz wireless communications, $L_\textrm {relative}(f)$ is more important than $L_\textrm {mode}(f)$ because the relative phase noise is transferred to the phase noise of the generated mm/THz waves. $L_\textrm {relative}(f)$ generated from the ECDL ($L_\textrm {relative,ECDL}(f)$) and DFB laser ($L_\textrm {relative,DFB}(f)$) is shown in Figs. 4(a) and (b). Note that, in the measurements, although we tried several combinations of the comb modes in the TWDI, the obtained results didn’t depend on the selected comb modes after calibrating at the comb mode spacing by dividing by $N^2$ (i.e., 540 GHz in this experiment). The relative phase noise is, in principle, free from the phase noise of the pump cw laser, resulting in the much better $L_\textrm {relative}(f)$ than $L_\textrm {mode}(f)$ when the DFB laser is used as a pump cw laser. As shown in Fig. 4(b), $L_\textrm {relative,DFB}(f)$ is about 30 dB better than $L_\textrm {mode,DFB}(f)$ for the wide range of the frequency offsets. Meanwhile, when the phase noise of the pump cw laser is comparative (or smaller) to the thermorefractive noise as in the case of the ECDL, $L_\textrm {relative,ECDL}(f)$ is similar to $L_\textrm {mode,ECDL}(f)$ for the 1st harmonic comb mode as shown in Fig. 4(a). Thus, the difference between $L_\textrm {relative,DFB}(f)$ and $L_\textrm {relative,ECDL}(f)$ is far smaller than the difference between $L_\textrm {mode,DFB}(f)$ and $L_\textrm {mode,ECDL}(f)$ as shown in Fig. 4(c). In addition, $L_\textrm {relative,DFB}(f)$ could be further reduced by setting the detuning between the frequency of the DFB laser and the resonance frequency at a quiet point, where the phase noise of the pump cw laser has the smallest influence on $L_\textrm {relative}(f)$ [15,16]. Therefore, the phase noise of the pump cw laser for the application of mm/THz wireless communications is not critical, but usability in addition to size, weight and power (SWaP) may be more important.
Fig. 4. (a) SSB phase noise PSD of $L_\textrm {mode,ECDL}(f)$ at the +1st harmonic (blue) and $L_\textrm {relative,ECDL}(f)$ (red). The wavelength of the ECDL is 1548 nm. (b) SSB phase noise PSD of $L_\textrm {mode,DFB}(f)$ at the +1st harmonic (blue) and $L_\textrm {relative,DFB}(f)$ (red). The wavelength of the DFB laser is 1548 nm. (c) Difference of the relative phase noise of the comb modes between the ECDL and DFB laser ($L_\textrm {relative,DFB}(f)$ - $L_\textrm {relative,ECDL}(f)$).
2.4 Limit of $L_\textrm {mode}(f)$ and $L_\textrm {relative}(f)$ by the intensity noise
As $L_\textrm {mode}(f)$ and $L_\textrm {relative}(f)$ can be limited by the intensity noise of microresonators with FSRs of a few tens of GHz, we also investigated for larger FSR microresonators. To investigate the influence, excess intensity noise is added to the pump cw laser by modulating the pump current of the LD of the EDFA. Unlike the modulation added for Fig. 2, here in this section, white intensity noise is added, although the responses of the Er-doped fiber and LD controller limit the bandwidth of the modulation. As shown in Fig. 5(a), the intensity noise of the ECDL ($L_\textrm {without,IM}$ (blue) and $L_\textrm {with,IM}$ (red) without and with the modulation, respectively) is increased by about 30 dB for a frequency offset from 100 Hz to 10 kHz, gradually decreasing the excess intensity noise for a frequency offset above 10 kHz. With the increased intensity noise, $L_\textrm {mode,ECDL}(f)$ and $L_\textrm {relative,ECDL}(f)$ shows the slight increase at a frequency offset between 1 kHz and 100 kHz as shown in Figs. 5(b) and (c). From these results, by subtracting the amount of the excess intensity noise (i.e., $L_\textrm {with,IM} - L_\textrm {without,IM}$), the upper limit of $L_\textrm {mode}(f)$ and $L_\textrm {relative}(f)$ of the microresonator set by the intensity noise of the pump cw laser (the blue curve in Fig. 5(a)) can be estimated as shown in the black curves in Figs. 5(b) and (c). Here, we didn’t plot the estimation for a frequency offset above 100 kHz, because the excess intensity noise was too small to observe the excess noise for $L_\textrm {mode,ECDL}(f)$ and $L_\textrm {relative,ECDL}(f)$. According to these results, the intensity noise of the pump cw laser with about −130 dBc/Hz and < 200 mW optical power as in the case of our ECDL is much smaller than the thermorefractive noise for the microresonators with large FSR. Advanced microresonators [26–28], which require less pump power to generate a soliton comb, would have even less sensitivity of the intensity noise.
Fig. 5. (a) Relative intensity noise (RIN) (left axis) and intensity noise (right axis) of the ECDL with (red) and without (blue) amplitude modulation. (b) $L_\textrm {mode,ECDL}(f)$ at the +1st harmonic with (red) and without (blue) amplitude modulation. (Black) Upper limit of the estimated thermorefractive noise for the comb mode set by the intensity noise. (c) $L_\textrm {relative,ECDL}(f)$ with (red) and without (blue) amplitude modulation. (Black) Upper limit of the estimated thermorefractive noise for the relative phase noise between the comb modes set by the intensity noise.
3. Conclusion
In conclusion, we investigated the phase noise of the comb modes and relative phase noise between the comb modes. When a pump cw laser with low phase noise (ECDL) was used, the thermorefractive noise was observed. In addition, the quadratic increase of the phase noise of the comb modes, as the comb mode number, counted from the frequency of the pump cw laser increases, was also observed, which is very different from conventional optical frequency combs based on mode-locked lasers. Meanwhile, when a pump cw laser with large phase noise (DFB laser) was used, the thermorefractive noise was not observed, showing the same phase noise of the comb modes, independent of the comb mode number. According to these observations, the phase noise of the comb mode can be expressed not only as $L_{n+k}(f) = (n+k)^2L_\textrm {rep}(f) + L_\textrm {ceo}(f)$, which is also true for the conventional frequency combs, but also as $L_{n+k}(f) \approx L_\textrm {cw}(f) + k^2L_\textrm {rep}(f)$. Here, $L_\textrm {rep}(f)$ is limited by the thermorefractive noise, as indicated in Ref. [18] and Ref. [19]. Regarding to the relative phase noise between the comb modes, the phase noise of the pump cw laser was not so critical, because the phase noise of the pump cw laser is cancelled out, in principle. Experimentally, $L_\textrm {relative,DFB}(f)$ was slightly larger than $L_\textrm {relative,ECDL}(f)$. We also showed that, unlike soliton combs with small comb spacing, the intensity noise of the pump cw laser was negligible.
For coherent optical communications, where the phase noise of the comb modes is important, the thermorefractive noise could be suppressed by coupling an auxiliary cw laser with blue-detuned wavelength against a resonance wavelength to a microresonator [18,29,30]. Regarding to the relative phase noise between the comb modes, the phase noise of the pump cw laser is not so critical, because the phase noise of the pump cw laser is mostly cancelled out. The cooling method [18,29,30] could also reduce the relative phase noise between the comb modes. Taking the advantages of the DFB laser in terms of usability and SWaP into consideration, the soliton comb generated from the DFB laser may be more suitable for the application of mm/THz wireless communications than the use of low phase noise pump cw lasers such as ECDLs.
Funding
Precursory Research for Embryonic Science and Technology (JPMJPR1905); Japan Society for the Promotion of Science (19H00871); Cabinet Office, Government of Japan (Subsidy for Reg. Univ.and Reg. Ind. Creation); Nakatani Foundation for Advancement of Measuring Technologies in Biomedical Engineering.
Disclosures
The authors declare no conflicts of interest.
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