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Abstract
Carrier phase of a harmonically mode-locked optical frequency comb (OFC) is investigated in detail. While harmonically mode-locked OFCs show promise for high-repetition-rate applications, their mode spacing is not the same as the pulse repetition rate, unlike fundamentally mode-locked OFCs. Consequently, harmonically mode-locked OFCs are unsuitable for applications requiring OFCs with wide mode spacing. This study examines the pulse-to-pulse carrier phase evolution of 4th- and 5th-order harmonically mode-locked OFCs, revealing uneven carrier phase evolution responsible for the narrow mode spacing. The possibility of achieving harmonically mode-locked OFCs with wide mode spacing is suggested by implementing pulse-to-pulse phase modulation to ensure even phase evolution.
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1. Introduction
Optical frequency combs (OFCs) of high repetition rates are extensively used for astrophysics [1,2], spectroscopy [3], and optical communications [4]. Electro-optic OFCs, Kerr OFCs, and microresonator OFCs are capable of achieving high repetition rates. While mode-locked laser OFCs exhibit excellent optical frequency stability and narrow spectral linewidth, their inherent repetition rates are relatively low (typically 20-200 MHz for Er-doped fiber laser OFCs). Therefore, for applications demanding repetition rates exceeding 1 GHz, various methods, such as the mode-filtering technique utilizing optical resonators [1–3,5–7], have been employed to increase the repetition rate of mode-locked OFCs. However, the mode-filtering process leads to substantial optical power reduction, sometimes necessitating optical amplification, which can introduce noise. Moreover, depending on the finesse of the filtering resonator, residual modes that are intended to be filtered out may persist in the processed OFC, potentially causing undesirable effects like mode frequency shifts after OFC mode bandwidth broadening using highly nonlinear fibers [8]. An alternative approach involves injection locking to multiply the mode spacing, but this method also has the problem of the persistence of residual irrelevant modes [9]. To overcome these difficulties, an interleaving technique has been introduced [10]. This technique theoretically does not incur optical power loss, and the intensity of irrelevant modes can be minimized to the level of detector noise. However, it comes at the cost of increased complexity and bulkiness in the setup.
As an alternative method to increase the repetition rate of mode-locked OFCs, the utilization of harmonically mode-locked lasers has been proposed [11,12]. In harmonic mode locking, multiple optical pulses circulate simultaneously within the laser cavity, resulting in a multiplied optical pulse repetition rate. However, it has been observed that the mode spacing of harmonically mode-locked OFCs (HML-OFCs) differs from the pulse repetition rate of HML-OFCs, and instead remains that of OFCs in fundamental mode locking (FML-OFCs), in which a single optical pulse circulates in the laser cavity [13,14]. Then while HML-OFCs can serve as high-repetition-rate OFCs for applications in which temporal regularity of the pulse envelope is important such as in microwave sources [15], they cannot be used for applications requiring wide mode spacing. The narrow mode spacing for HML-OFCs arises from uneven pulse phase evolution of the carrier wave among the multiple pulses within the cavity [16]. A similar phenomenon is observed in the interleaving scheme [10], where carrier phase shifts associated with reflection result in uneven carrier phase evolution, causing a mismatch between mode spacing and pulse repetition rate. In previous work [10], pulse-to-pulse phase modulation successfully compensated for uneven phase evolution, enabling mode spacing multiplication. Similarly, the application of phase demodulation techniques to HML-OFCs is anticipated, and to achieve this, a thorough investigation into the phase structure of HML-OFCs is essential. The study on the cross-correlation between successive pulses has been carried out [17]. In this study, we study the carrier phase structure of HML-OFCs using an alternative approach, namely, examining the interferogram between the HML-OFC and a single-mode continuous-wave (cw) laser in time domain. The results reveal uneven carrier phase evolution of the pulses in the laser cavity, making the mode spacing of HML-OFCs the same as that of FML-OFCs. Then it is suggested that for the wide mode spacing of HML-OFCs, a suitable phase modulation function to compensate for the observed carrier phase shift is applied. Additionally, our study suggests the presence of phase-correlated interactions among the circulating pulses.
2. Carrier phase evolution of the HML-OFCs
The mathematical expression for the HML-OFCs has previously been given in [16], but for the convenience of the readers, it will be provided in this section. In $n$th-order HML-OFCs ($n$-HML-OFCs), the laser cavity contains $n$ pulses circulating within it. The output pulse originating from the same pulse circulating in the laser cavity is ejected from the cavity every $n$th time. This is illustrated schematically in Fig. 1 for the case of $n=4$. In Fig. 1, $j$ and $k$ denote the circulation number of the pulses in the cavity and the order of the pulse in the cavity, respectively. The output pulse is identified with $j$ and $k$. Output pulses that share the same value of $k$ originate from the $k$th pulse in the cavity, making them phase-correlated. Consequently, the carrier phase of the ($nj+k$)th pulse in the output beam can be expressed as,
where $\varphi _{nj+k}^{(n)}$ represents the carrier phase relative to the pulse envelope center of the $n$-HML-OFC, $\phi _k^{(n)}$ the carrier phase of the $k$th circulating pulse in the cavity with respect to the pulse envelope center for $j=0$, and $\phi _c$ is an offset phase shift experienced by the pulse after circulating once in the laser cavity [it is the carrier-envelope offset phase for the FML-OFC ($n=1$)].
Fig. 1. A schematic illustrating the optical pulses circulating within the laser cavity and the resulting output pulses from the 4th-order HML-OFC. Each output pulse is characterized by two integers, $j$ and $k$, where $j$ represents the pulse’s circulation number within the cavity, and $k$ denotes the order of the circulating pulse in the cavity.
The electric field of the FML-OFC is expressed as (in complex expression),
The frequency-domain distribution of $E^{(n)}(t)$ in Eq. (4) is represented as the product of the Fourier transform of $f_M^{(n)}(t)$ and $E_c(t)$. The Fourier transform of $f_M^{(n)}(t)$ is given by,
The Fourier transform of $E_c(t)$ is well-known as,
As a result, the Fourier transform of $E^{(n)}(t)$ in Eq. (3) is expressed as,
Here, $\Phi _q^{(n)}$ represents the discrete Fourier coefficient as,
The mode number $m$ in Eq. (7) is substituted with $np+q$ in Eq. (8), and the periodicity of $\Phi _q^{(n)}$, namely, $\Phi _{q+n}^{(n)}=\Phi _q^{(n)}$, is applied. From Eq. (8), it is evident that the mode intensity is determined by $\left |\Phi _q^{(n)}\right |^2$ and the broadband spectral distribution of $G(\omega -\omega _0)$. Equations (8) and (9) suggest that the mode spacing of the HML-OFC is $\omega _{\rm rep}^{(1)}$, assuming every value of $\Phi _q^{(n)}$ is non-zero. If the phase of multiple pulses in the laser cavity evolves evenly, namely,
then $\Phi _q^{(n)}$ is zero for $q\neq n'$ ($n'$ is an integer between 0 and $n-1$) and $n$ for $q=n'$. Consequently, for achieving a wide mode spacing of $\omega _{\rm rep}^{(n)}(=n\omega _{\rm rep}^{(1)})$ with the $n$-HML-OFC, the phase evolution from pulse to pulse of the circulating laser pulses should evolve evenly as Eq. (10). Otherwise, if the phase evolution is uneven, the mode spacing of the HML-OFC remains unchanged from that of the FML-OFC (the mode intensity depends on the amplitude of $\Phi _q^{(n)}$).In this analysis, the determination of the carrier phase structure of the circulating pulses in the laser cavity, denoted as $\phi _k^{(n)}$, is crucial for understanding the mode spacing and the distribution of mode intensity. Experimentally, the phase of each pulse can be investigated through heterodyne detection using a cw laser, with which the time-domain interference signal (an interferogram) between the OFC and the cw laser is acquired by an analog-digital converter.
3. Experiment and discussions
To analyze the mode spacing and the carrier phase of the HML-OFC, we observe the heterodyne signal between the HML-OFC and a cw laser. Our setup involves an OFC of a Yb-doped fiber mode-locked laser using nonlinear polarization rotation, as illustrated in Fig. 2(a). The pulse repetition rate under the fundamental mode locking condition $\omega _{\rm rep}^{(1)}/2\pi$ is 66.7 MHz. Initially, the fundamental mode locking is achieved, and subsequently, transition to harmonic mode locking condition occurs by precise adjustments, such as the waveplate rotation in the experimental setup. Once harmonic mode locking is established, it can be maintained for an extended period (over 1 day). The pulse train of the 4th-order HML-OFC (4-HML-OFC) is shown in Fig. 3(a), and the corresponding rf spectrum is given in Fig. 3(b), alongside the results of the FML-OFC for comparison. The pulse repetition rate of the 4-HML-OFC ($\omega _{\rm rep}^{(4)}/2\pi$) is indeed quadrupled (267 MHz) with suppression of supermode noise ($\sim$60 dB). Both the FML-OFC and the 4-HML-OFC exhibit broadband optical spectra, as shown in Fig. 3(c).
Fig. 2. (a) Experimental setup of the Yb-doped fiber OFC mode-locked by nonlinear polarization rotation. (b) Configuration for the beat-note measurement between the OFC of (a) and a single-mode cw laser. Abbreviations: M for mirrors, G for diffraction gratings, PBS for a polarized beam splitter, HWP for half-wave plates, QWP for a quarter-wave plate, AL for aspheric lenses, WDM for a wavelength division multiplexer, OI for an optical isolator, BPD for balanced photodetectors, and FC for a 50/50 fiber coupler.
Fig. 3. Optical intensity comparison of the FML-OFC and the 4-HML-OFC in both time domain (a) and frequency domain (b). In the time domain, the peak power of the 4-HML-OFC pulses is 1/4th that of the FML-OFC, resulting the identical average power. The supermode suppression in the 4-HML-OFC in (b) is less than 60 dB. Optical spectra of the FML-OFC and the 4-HML-OFC are illustrated in (c).
To measure the mode spacing, we employ heterodyne detection of the OFC through mixing with a single-mode cw laser. The resulting beat-note spectrum is observed by a spectrum analyzer, as illustrated in Fig. 2(b). The cw laser is a handmade grating-feedback external cavity laser diode operating at 1030 nm. In Fig. 4(a), the beat-note spectra of the FML-OFC, 4-HML-OFC, and 5-HML-OFC with the cw laser are given. The mode spacing of the HML-OFC turns out to be the same as that of the FML-OFC, with no multiplication of the mode spacing. The similar results are observed for 2-HML-OFC and 3-HML-OFC in [13,14], where the suggested reason is the unleven phase evolution from pulse to pulse, consistent with the conclusion drawn in the previous section.
Fig. 4. (a): Beat-note spectra of the FML-OFC, 4-HML-OFC, and 5-HML-OFC with a 1030 nm single-mode cw laser. Vertical dotted lines indicate the repetition rate of the FML-OFC (66.7 MHz) and its harmonics. The spectral lines at the dotted lines emerge due to the imperfect balance of the 50/50 fiber coupler and the balanced photodetectors in Fig. 2(b). (b): Portions of interferogram for the FML-OFC, 4-HML-OFC, and 5-HML-OFC with a 1030 nm single-mode cw laser. In the result of the 4-HML-OFC, the time segments corresponding to the value of $k$ are indicated by arrows. The segments of $k$ contain a pulse train originating from a $k$th single pulse circulating in the laser cavity. Such a pulse train can be obtained by multiplying $w_k^{(n)}$ in Eq. (11).
For thorough analysis aimed at evaluating the carrier phase $\phi _k^{(n)}$ of the circulating pulses, the time-domain interference signal between the OFC and the cw laser (interferogram) is observed using a high-speed oscilloscope (Tektronix, DPO7104, bandwidth is 1 GHz), as depicted in Fig. 4(b). The interferograms are recorded at a 50 ps interval for a duration of 10 $\mu$s ($2\times 10^5$ points). Over this short period, laser frequency drift is considered negligible. To extract $\phi _k^{(n)}$ from the results in Fig. 4(b), a window function of a square pulse wave function with a duty cycle of $1/n$,
Equation (13) suggests that the Fourier component phase is proportional to the mode number $m$, and the offset value of $\theta _{m,k}^{(n)}$ corresponds to $\phi _k^{(n)}$. Therefore, $\phi _k^{(n)}$ is determined by finding $\theta _{m,k}^{(n)}$ and analyzing its dependence on $m$ and $k$.
Figure 5(a) shows the dependence of the phase of the Fourier component of the extracted interferogram ($\theta _{m,k}^{(4)}$) on the mode number $m$ and the pulse number $k$ for the 4-HML-OFC. In the figure, the mode number is given as a difference from $m_0$, which is the number of the OFC mode closest to the cw laser frequency, and the phase is unwrapped to highlight the linear dependence on $m$ clearly, taking into account the $2\pi$-periodicity of the phase. As anticipated in Eq. (13), linear dependence of $\theta _{m,k}^{(4)}$ on $m$ is confirmed, and the slope is the function of $k$. In Fig. 5(b), the value of $\theta _{m,k}^{(4)}$ for $m=m_0$ is presented as a function of $k$. According to Eq. (13), any deviation from the linear dependence in Fig. 5(b) is due to $k$-dependence of $\phi _k^{(4)}$. For the case of the 4-HML-OFC, a noticeable deviation from linear dependence on $k$ is observed at $k=2$, implying that $\phi _2^{(4)}$ is approximately $\pi$, while $\phi _{0,1,3}^{(4)}$ are defined as zero. This result confirms that the pulse-to-pulse carrier phase of the HML-OFC evolves unevenly, maintaining the mode spacing unchanged from that of the FML-OFC. In fact, the absolute values of $\Phi _q^{(4)}$ in Eq. (9), corresponding to the mode intensity, are calculated as 2 for every $q$, consistent with the results in Fig. 4(a). Similarly for the 5-HML-OFC, the phase of the Fourier component of the extracted interferogram ($\theta _{m,k}^{(5)}$) linearly depends on the mode number $m$ [Fig. 5(c)], and the mode phase for $m=m_0$ exhibits linear dependence on $k$ except for $k=0$, resulting in $\phi _0^{(5)}\sim \pi /2$ and $\phi _{1,2,3,4}^{(5)}=0$ [Fig. 5(d)]. The observed phase evolution results in an unchanged mode spacing from the FML-OFC as well, as the absolute values of $\Phi _q^{(5)}$ are $\sqrt {17}$ for $q=0$ and $\sqrt {2}$ for others. Thus, the mode intensity distribution of the 5-HML-OFC is anticipated not to be even, in contrast to the 4-HML-OFC. It is noted that quantitative discussion of the mode intensity distribution observed in Fig. 4(a) is difficult due to the inherent frequency dependence of the rf response of the balanced photodetectors.
Fig. 5. (a, c): Unwrapped phases of the Fourier components of the extracted pulse train interferograms for the 4-HML-OFC (a) and the 5-HML-OFC (c) as a function of the mode number $m$. The index $m_0$ corresponds to the OFC mode closest to the cw laser frequency. Dotted lines represent the results of linear fitting. (b, d): Phases of the Fourier components for $m=m_0$ of the 4-HML-OFC (b) and the 5-HML-OFC (d) as a function of the pulse index circulating in the laser cavity $k$. The linear lines indicate the results of linear fitting for 3 points in (b) and for 4 points in (d). In (d), the phase periodicity of $2\pi$ is considered.
In this observation, it is confirmed that the mode spacing of the HML-OFC is not identical with its pulse repetition rate due to uneven carrier phase evolution. Consequently, HML-OFCs cannot work as wide mode-spacing OFCs, even with a high pulse repetition rate. When employing the interleaving technique to multiply the mode spacing, a similar carrier phase modulation occurs, and correspondingly, the mode spacing is not multiplied [10]. In the case of the interleaving, however, successful multiplication of the mode spacing has been achieved by employing pulse-to-pulse phase (de)modulation using an electro-optic phase modulator to compensate for uneven phase evolution. Therefore, it is anticipated that, even in the case of HML-OFCs, the mode spacing can be multiplied by implementing pulse-to-pulse phase demodulation using an electro-optic phase modulator as well. For the phase demodulation, the pulse repetition rate must be stabilized, since the demodulation signal applied to the modulator must synchronize with the optical pulse.
The fact of uneven carrier phase evolution of the HML-OFC suggests that the carrier phases of the optical pulses circulating in the laser cavity of the HML-OFC become mutually fixed. This implies the presence of some coherent interaction among the pulses within the cavity. In [18], it is stated that the acoustic effect in the optical fiber can lead to regularly spaced pulse trains in the cavity. Since the sound in the fiber is induced by the pulse envelope, the acoustic effect is presumed to be independent of the carrier phase of the pulse. Therefore, for a comprehensive and quantitative understanding of HML-OFCs, a theoretical discussion is essential to the carrier phase correlation.
In the observation of the beat-note spectra [Fig. 4(a)], while the beat-note spectral lines drift in frequency due to the frequency drift of the cw laser and the OFC mode frequency, the intensity of the spectral lines seems to be stable. Therefore, it is expected that the phase correlation among the pulses in the cavity is also stable, because the mutual phases among the pulses in the cavity determine the beat-note intensity [Eq. (9)]. However, it is also possible that the mutual phases drift with maintaining the constant OFC mode intensities, and therefore confirmation of the phase correlation for a longer period of time than the recording time in this study ($\sim 10 \mu$s) is necessary. For this purpose, the beat-note frequency drift must be suppressed by phase-locking the cw laser with respect to one of the OFC mode frequency.
4. Conclusions
We investigate higher-order HML-OFC in pursuit of wide mode spacing OFCs. Utilizing a ytterbium-doped fiber laser at 1030 nm, we have generated 4th- and 5th-order HML-OFCs. The mode spacing is evaluated by the heterodyne detection with a single-mode cw laser. The results indicate that, despite succsessfully multiplying the pulse repetition rate, the mode spacing of the HML-OFCs remains unchanged compared to that of the FML-OFC. A detailed interferogram analysis reveals that the unsuccessful mode-spacing multiplication is due to the uneven phase evolution. This result suggests the existence of coherent interaction among the pulses circulating in the laser cavity to establish phase correlation between pulses. The long-term carrier phase measurement of the HML-OFCs is anticipated to investigate stability of the phase correlation between the circulating pulses in the laser cavity and dependence of the phase correlation on the OFC parameters such as the pulse repetition rate and the carrier-envelope offset frequency. To achieve a wide mode spacing OFC, phase compensation for uneven phase evolution using an electro-optic phase modulator is proposed, similar to the approach demonstrated in the case of the mode-spacing multiplication by pulse interleaving [10].
Funding
Japan Society for the Promotion of Science (21K04930).
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available currently but may be obtained from the authors upon reasonable request.
References
1. T. Steinmetz, T. Wilken, C. Araujo-Hauck, et al., “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008). [CrossRef]
2. T. Wilken, G. Lo Curto, R. A. Probst, et al., “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012). [CrossRef]
3. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007). [CrossRef]
4. J. Pfeifle, V. Brasch, M. Lauermann, et al., “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014). [CrossRef]
5. M. S. Kirchner, D. A. Braje, T. M. Fortier, et al., “Generation of 20 ghz, sub-40 fs pulses at 960 nm via repetition-rate multiplication,” Opt. Lett. 34(7), 872–874 (2009). [CrossRef]
6. T. Hasegawa, “Broadened optical spectrum of repetition-rate-multiplied erbium-doped fiber comb,” J. Opt. Soc. Am. B 33(10), 2057–2061 (2016). [CrossRef]
7. J. Lee, S.-W. Kim, and Y.-J. Kim, “Repetition rate multiplication of femtosecond light pulses using a phase-locked all-pass fiber resonator,” Opt. Express 23(8), 10117–10125 (2015). [CrossRef]
8. G. Chang, C.-H. Li, D. F. Phillips, et al., “Toward a broadband astro-comb: effects of nonlinear spectral broadening in optical fibers,” Opt. Express 18(12), 12736–12747 (2010). [CrossRef]
9. C.-G. Jeon, S. Zhang, J. Shin, et al., “Highly tunable repetition-rate multiplication of mode-locked lasers using all-fibre harmonic injection locking,” Sci. Rep. 8(1), 13875 (2018). [CrossRef]
10. T. Kageyama and T. Hasegawa, “Mode spacing multiplication of optical frequency combs without power loss,” Opt. Express 30(11), 19090–19099 (2022). [CrossRef]
11. B. Ortaç, A. Hideur, and M. Brunel, “Passive harmonic mode locking with a high-power ytterbium-doped double-clad fiber laser,” Opt. Lett. 29(17), 1995–1997 (2004). [CrossRef]
12. J. J. McFerran, “Échelle spectrograph calibration with a frequency comb based on a harmonically mode-locked fiber laser: a proposal,” Appl. Opt. 48(14), 2752–2759 (2009). [CrossRef]
13. S. Herr, T. Steinmetz, T. Wilken, et al., “Optical mode structure of a harmonically mode-locked yb femtosecond fiber laser,” in CLEO:2011 - Laser Applications to Photonic Applications, (Optica Publishing Group, 2011), p. CMS3.
14. M. Yan, W. Li, K. Yang, et al., “Harmonic mode locking with reduced carrier-envelope phase noise in ytterbium-doped fiber laser,” Opt. Lett. 37(15), 3021–3023 (2012). [CrossRef]
15. A. Bartels, S. A. Diddams, C. W. Oates, et al., “Femtosecond-laser-based synthesis of ultrastable microwave signals from optical frequency references,” Opt. Lett. 30(6), 667–669 (2005). [CrossRef]
16. F. Quinlan, S. Ozharar, S. Gee, et al., “Harmonically mode-locked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11(10), 103001 (2009). [CrossRef]
17. K. S. Abedin, J. T. Gopinath, L. A. Jiang, et al., “Self-stabilized passive, harmonically mode-locked stretched-pulse erbium fiber ring laser,” Opt. Lett. 27(20), 1758–1760 (2002). [CrossRef]
18. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, “Acoustic effect in passively mode-locked fiber ring lasers,” Opt. Lett. 20(8), 907–909 (1995). [CrossRef]
