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Abstract
We demonstrate the mode spacing multiplication of optical frequency combs (OFCs) using interleaving technique. The pulse train of an OFC after interleaving is phase modulated, and phase demodulation is necessary for mode spacing multiplication. This scheme enables mode spacing multiplication without power loss in principle, which is in contrast to the conventional mode filtering technique. During demonstration, the mode spacing of the OFC of a mode-locked Er-doped fiber laser (repetition rate of 97 MHz) is quadrupled by the interleaving process and successive phase demodulation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The repetition rate of optical frequency combs (OFCs) determines frequency mode spacing, and therefore, it is one of the most important parameters, especially for applications in which the equally spaced mode distribution in frequency domain of OFCs is used. In the case of the OFCs of mode-locked Er-doped fiber lasers, the repetition rate is typically between 20 and 200 MHz. With regard to the application of OFCs to astrophysics [1,2], spectroscopy [3], and optical communications [4], OFCs of higher mode spacing are required. To obtain such OFCs, mode-filtering technique by an optical cavity is extensively employed [5–8]. The OFCs of highly modulated continuous-wave lasers [9] can be directly used for these applications; however, the high spectral stability and narrow spectral linewidth of the OFCs of mode-locked lasers are beneficial for precise measurement. An OFC of a mode-locked titanium sapphire laser with a special cavity design of 10 GHz repetition rate is also suitable for these applications in the wavelength region of 800 nm [10]. With the filtering technique, by making the free spectral range of the cavity be integer ($N$) multiple of the mode spacing of the OFC, the mode spacing of the transmitted beam is expected to be multiplied by $N$. In this process, however, the optical average power of the OFC is reduced down to $1/N$. With optical amplification after mode filtering, the total power can be recovered, but noise level increases simultaneously. In addition, depending on the finesse of the filtering cavity, the transmitted beam includes a portion of OFC modes that should be removed (irrelevant modes). These irrelevant modes may cause unexpected behaviors, such as the mode frequency shift after OFC mode bandwidth broadening using a highly nonlinear fiber [11]. Another scheme to multiply the mode spacing by injection locking has been reported, in which the problem of residual irrelevant modes exists [12].
We introduce a new scheme to multiply the OFC repetition rate (mode spacing) without optical power loss. In this scheme, an optical delay line (a pulse interleaver) is used to interleave a pulse between two consecutive pulses in time domain to double the pulse repetition rate. Pulse interleavers are introduced as a radio-frequency (rf) doubler to use OFCs as low-noise microwave generators [13,14]. Although the pulse interleaver doubles the pulse envelope repetition rate, the mode spacing of the OFC remains unchanged because the carrier phase is modulated as a consequence of interleaving, as explained later. In general, the mode spacing of OFCs is identical with the pulse envelope repetition rate, while the carrier phase of the pulse train evolves constantly (this carrier phase evolution during the pulse interval is defined as the carrier-envelope offset phase). However, in the case of the phase-modulated OFCs, the carrier phase evolution is not constant, and the optical mode distribution of narrower spacing than the pulse envelope repetition rate is obtained [15–19]. For low-noise microwave generation [13,14] and multiphoton imaging [20] as the applications of high-repetition-rate OFCs, the periodicity of the pulse envelope of the OFCs plays an important role, and the carrier phase is considered to be irrelevant. However, if the equally-spaced frequency-domain mode distribution of OFCs is essential (such as in astrophysics, spectroscopy, and optical communications), the carrier phase demodulation after interleaving is necessary. In [21], mode spacing multiplication using on-chip interleavers has been investigated without phase demodulation; however, as in the optical cavity filtering technique, the optical power is reduced down to $1/N$.
In this study, carrier phase demodulation using a wave-guide electro-optic modulator (WG-EOM) is performed to multiply mode spacing without power loss. As a proof-of-principle experiment, the repetition rate (mode spacing, $f_{\textrm{rep}}=97$ MHz) of the mode-locked Er-doped fiber laser OFC at 1550 nm is multiplied by a factor of 4. The mode distribution of the interleaved OFC in frequency domain is confirmed by beat-note measurements with a single-mode continuous-wave (cw) laser. The mode spacing is indeed quadrupled, and in addition, the spectral intensity of irrelevant modes (at $f_{\textrm{rep}}$, $2f_{\textrm{rep}}$, and $3f_{\textrm{rep}}$) is substantially suppressed compared with that of the mode-spacing multiplied OFCs based on optical cavity filtering and injection locking. In this study, the mode spacing of the OFC should be distinguished from the pulse envelope repetition rate because their values are not always the same.
2. Free-space interleavers
First, we observe the pulse repetition rate of an OFC after passing through free-space interleavers. Although fiber interleavers have been used in [13,14], the path length adjustability and small refractive index dispersion of free-space interleavers are beneficial. The experimental setup is shown in Fig. 1. As an OFC, we use an Er-doped fiber laser of nonlinear polarization rotation mode locking. The repetition rate ($f_{\textrm{rep}}$) and the carrier-envelope offset frequency ($f_{\textrm{CEO}}$) are stabilized at 97 MHz and 30 MHz, respectively (details are not shown in Fig. 1). The OFC beam is polarized by a polarizer and divided into two beams with 50/50 by a polarized beam splitter in a free-space interleaver. The longer path length in the interleaver is 1.5 m longer than the shorter one. The pulse propagating in the longer path delays for $\sim 5$ ns ($=1/2f_{\textrm{rep}}$) with respect to the pulse in the shorter path, and after combining them at the polarized beam splitter, the pulse envelope repetition rate is doubled with alternating polarization. Similarly, the second interleaver doubles the pulse envelope repetition rate, and finally, the repetition rate is quadrupled. The interleaved OFC beam is detected by a photodetector (5 GHz bandwidth, DET08CFC/M, Thorlabs) and monitored by an oscilloscope (1 GHz, Tektronix DPO7104) or a rf spectrum analyzer (6 GHz, Tektronix MDO4024C with an optional spectrum analyzer). Here, the alternating polarization of the pulse train does not affect the result owing to polarization insensitivity of the photodetector [14]. It is emphasized that $f_{\textrm{rep}}$ is defined as the repetition rate (or the mode spacing) of the original OFC and not that of the interleaved OFC.
Fig. 1. Experimental setup for pulse envelope repetition rate multiplication with free-space interleavers. The repetition rate and carrier-envelope offset frequency of the OFC are stabilized (not shown in the figure). The input pulse train in the first interleaver is linearly polarized to 45 degrees to divide into two pulse trains at the first polarized beam splitter (PBS) in the first interleaver. The output of the first interleaver constitutes the optical pulse train of alternating polarization (horizontal and vertical) with the repetition rate of $2f_{\textrm{rep}}$ after the coupling at the second PBS. The half-wave plate (HWP) between the two interleavers rotates the polarization direction by 45 degrees to divide the pulse train into two at the first PBS in the second interleaver. Abbreviations: CL stands for collimator lens, TS for translation stages, and PD for photodetector.
The time-domain result is shown in Fig. 2(a), in which the pulse trains without and with interleaving are denoted with black and red lines, respectively. The pulse envelope repetition rate is quadrupled with interleaving. The peak intensity of the interleaved pulse train is 1/4 of the original pulse peak intensity, so that the average power of the pulse train is conserved in the interleaving process. The spectrum of the interleaved pulse train is shown in Fig. 2(b). Spectral lines at 384 MHz ($4f_{\textrm{rep}}$) and its second harmonic at 768 MHz are observed, whereas the irrelevant lines (spectral lines expected to be removed, at 97 MHz etc.) are less than the noise level. This result implies that the side-mode suppression ratio is > 63 dB.
Fig. 2. Results of the pulse envelope repetition rate measurement. In (a), time-domain optical intensity is shown (black and red lines represent pulse trains without and with interleaving, respectively). The optical intensity is normalized with respect to the peak intensity of the pulse train without interleaving. In (b) and (c), the rf spectra of the interleaved OFC are shown (resolution bandwidth is 50 kHz). The optical path length of the second interleaver for (c) is 1.4 mm longer than that for (b). In (d), the spectral line intensity at $2f_{\textrm{rep}}$ (blue), $4f_{\textrm{rep}}$ (red), and $6f_{\textrm{rep}}$ (green) is shown as a function of the delay time error (lower horizontal axis) and the optical path length difference (upper axis). The yellow area represents the noise level.
When the delay length of the second interleaver is 1.4 mm longer than that of the best case [Fig. 2(b)], irrelevant spectral lines exceed the noise level [Fig. 2(c)]. In Fig. 2(d), the line intensity is shown as a function of the delay time error of the second interleaver, and it is found that the irrelevant rf mode intensity is less than the noise level when the delay length error is < 100 $\mu$m. For OFCs with higher repetition rate, the phase error induced by the path length error becomes relatively substantial, which may practically determine the highest multiplication factor available with the interleaving technique (to be discussed later).
As mentioned before, the polarization of the interleaved OFC is alternating from pulse to pulse. The polarization can be aligned using the fast polarization control technique [22]. The polarization alignment setup is shown in Fig. 3(a). Time-domain optical intensity of the interleaved pulse train with and without the fast polarization control is shown in Fig. 3(b). Without the polarization alignment, the pulse polarization alternates, and therefore, every other pulse passes through the polarizer in front of the photodetector. As a result, the pulse repetition rate becomes $2f_{\textrm{rep}}$. With the polarization alignment, every pulse passes through the polarizer, resulting in the pulse repetition rate of $4f_{\textrm{rep}}$. In addition, the peak intensity of each pulse is unchanged with the fast polarization control, and therefore it is confirmed that the polarization-aligned optical pulse train can be obtained without power loss. Although the pulse envelope repetition rate is successfully multiplied without power loss, we do not use the fast polarization control technique hereafter in this study because of the simplicity of the experimental setup, and alternatively, a polarized beam splitter is used for the polarization alignment at the expense of the half of the optical power.
Fig. 3. Setup of the polarization alignment without power loss (a) and time-domain optical intensity of the interleaved pulse train without (black) and with (red) the fast polarization control (b). The details of the fast polarization control are described in [22].
3. Mode spacing multiplication
To examine the mode spacing of the OFC after the pulse envelope repetition rate multiplication, the beat-note spectrum between the OFC and the single-mode cw laser is observed as follows [Fig. 4(a)]. The cw laser is a grating-feedback external-cavity laser diode at 1560 nm, and the laser frequency is stabilized with respect to one of the OFC modes before interleaving [23]. The cw laser beam is combined with the interleaved OFC beam using a 50/50 optical splitter. The beat-note signal is obtained using balanced photodetectors (Thorlabs, PDB480C-AC) and monitored using the rf spectrum analyzer. The beat-note spectra without and with interleaving are illustrated in Fig. 5(a) and Fig. 5(b), respectively. Without interleaving [Fig. 5(a)], the spectral lines at $f_{\textrm{rep}}$ and its harmonics are observed, whereas with interleaving [Fig. 5(b)], the spectral lines except $4f_{\textrm{rep}}$ and $8f_{\textrm{rep}}$ are reduced down to a level less than the noise level. This result implies that the pulse envelope repetition rate multiplication is successfully carried out. Ideally, the spectral lines at the repetition rate and its harmonics should not be observed by the balanced detection [24], but practically they are observed due to the small unbalance of the 50/50 optical splitter and the balanced photodetectors.
Fig. 4. Measurement setup of the beat-note spectrum between the OFC and the single-mode cw laser without (a) and with (b) the carrier phase demodulation. The cw laser is stabilized with respect to one of the OFC modes before interleaving. BPD and AFG stand for balanced photodetectors and an arbitrary function generator, respectively.
Fig. 5. Beat-note spectra between the OFC and the cw laser. (a) is for the OFC without interleaving, and (b) is for the interleaved OFC. (c) is for the OFC after interleaving and phase demodulation. The spectral lines at $f_{\textrm{rep}}$ and its multiples in (a) and at $4f_{\textrm{rep}}$ and $8f_{\textrm{rep}}$ in (b) and (c) correspond to the pulse envelope repetition rate. Other lines are the beat-note spectrum lines. In (a), the reduction in the beat-note spectral line intensity in the high frequency region may be due to saturation of a rf amplifier in the balanced photodetectors.
Other spectral lines shown in Figs. 5(a) and (b) correspond to the beat note between the OFC and the cw laser. The beat-note spectrum remains unchanged even with inverleaving, namely, the mode spacing is not multiplied by interleaving. The reason is based on the OFC carrier phase modulation as follows.
Here, the optical electric field of the OFC at A-F specified in Fig. 4(a) is examined. It is assumed that the carrier-envelope offset phase of the original OFC ($\phi _{\textrm{CEO}}=2\pi f_{\textrm{CEO}}/f_{\textrm{rep}}$) is zero. At A in Fig. 4(a), the electric field of a single pulse of the OFC is expressed as,
After the second interleaver, the vertical component is again delayed by $t_{\textrm{d2}}$ in the second interleaver, and the electric field at D is described as,
If $\phi _{\textrm{int1}}$ and $\phi _{\textrm{int2}}$ are integer-multiples of $2\pi$, the first term in Eq. (5) is out of phase with respect to the second, third, and fourth terms. The electric field corresponding to the fifth pulse is the same as the first term in Eq. (5) in the case of $\phi _{\textrm{CEO}}=0$. Therefore, the $(4n+1)$-th pulse is phase-shifted by $\pi$ compared with other pulses ($n$ is an integer), and consequently, the pulse train is phase modulated. As mentioned in the section of Introduction, the mode spacing of OFCs is identical with the pulse envelope repetition rate when the carrier phase of the pulse train evolves constantly ($\phi _{\textrm{CEO}}$ for normal OFCs). However, the carrier phase evolution of the phase modulated OFCs is not constant, and the optical mode distribution of narrower spacing than the pulse envelope repetition rate is obtained [15–19]. The detailed discussion of the mode distribution of the phase-modulated OFCs is given in [18]. Hence, to obtain the pulse train of mode spacing identical to the pulse envelope repetition rate, the carrier phase must be demodulated for the constant carrier phase evolution.
For the carrier phase demodulation, we use a WG-EOM (EO Space, PM-0S5-10-PFU-PFU), which operates at low voltage with fast response ($V_{\pi }=$ 4.5 V, rf bandwidth of 10 GHz). The experimental setup is shown in Fig. 4(b). After interleaving and the polarization alignment by a polarized beam splitter, the pulse train is demodulated by the WG-EOM. The demodulation voltage from an arbitrary function generator to the WG-EOM is sinusoidal at $f_{\textrm{rep}}=97$ MHz and 1.9 V$_{\textrm{0}-{\textrm{p}}}$, which corresponds to $\pi /2$ phase shift amplitude (the disagreement with the catalog value may be due to the rf transfer function of the WG-EOM). The sinusoidal voltage is phase-locked with the local oscillator to stabilize the repetition rate of the original OFC. By choosing the phase of the demodulation voltage, the additional phase applied to the first, second, third, and fourth optical pulses in Eq. (5) of the interleaved OFC is $0$, $-\pi /2$, $0$, and $\pi /2$, respectively, and the total carrier phase is $0$, $\pi /2$, $\pi$, and $3\pi /2$, respectively, to realize constantly evolving carrier phase pulses. It is noted that the sinusoidal demodulation voltage cannot be used for mode spacing multiplication of a factor of > 8.
The OFC mode distribution is observed by measuring the beat-note spectrum with the cw laser. The result is shown in Fig. 5(c), in which two beat-note spectral lines between 0 and $4f_{\textrm{rep}}$ are observed, and irrelevant beat-note lines are now less than the noise level, which is determined by the intensity noise of the OFC. Therefore, it is confirmed that the mode spacing is indeed quadrupled. In addition, the beat-note spectral line intensity in Fig. 5(c) is increased by 6 dB compared with that without the demodulation [Fig. 5(b)], and meanwhile, the noise level is unchanged. This result is a consequence of the total optical power conservation and reduction of the OFC mode number. It is confirmed that interleaving does not degrade the beat-note spectral linewidth (< 1 Hz, which is determined by the resolution bandwidth of the rf spectrum analyzer).
In the aforementioned discussion, it is assumed that $\phi _{\textrm{int1}}$ and $\phi _{\textrm{int2}}$ are the integer multiples of $2\pi$, and $\phi _{\textrm{CEO}}$ is zero. In the case of non-zero $\phi _{\textrm{CEO}}$, mode spacing is multiplied if one of the conditions of
As $\phi _{\textrm{int}1}$ and $\phi _{\textrm{int}2}$ are determined by the path length difference of the interleavers, they drift owing to mechanical vibration and temperature drift. Therefore, for the stable operation of the mode spacing multiplication, the path difference in the interleavers must be stabilized within the carrier wavelength. Without stabilization, the irrelevant beat-note spectral lines appear a few seconds after the result shown in Fig. 5(c) is obtained.
The interleaver stabilization setup is shown in Fig. 6. As a reference of the path length difference, the cw laser for the beat-note measurement is used. The cw laser beam path is transversely displaced from the OFC path in the interleaver. The interference fringe of the cw laser between the short and long paths in the interleaver can be used as a feedback error signal for path length stabilization. The path length correction is supposed to be within a few wavelengths (1550 nm), which is in the acceptable range with regard to the path length error for pulse envelope repetition rate multiplication [Fig. 2(d)]. To close the feedback loop, the error signal is fed back to the piezo-electric transducer on which a mirror in the interleaver is attached. With the path stabilization for the two interleavers, the mode spacing multiplication is kept for hours [in fact, the spectrum in Fig. 5(c) is obtained with the path length stabilization].
Fig. 6. Setup of the interleaver with path length stabilization. The cw laser polarization is perpendicular to that of the OFC. The OFC (cw laser) beam path is denoted by the red solid (dotted) line. The cw laser path is transversely displaced from the OFC path. The interference of the cw laser between the short and long paths in the interleaver is detected at the photodetector (PD), and the feedback loop is to fix the optical intensity. PZT stands for a piezoelectric transducer. The servo circuit consists of a comparator with dc voltage to determine lock point, an amplifier, and electric filters.
The path length stabilization is separately required for every interleaver, and this complicates the whole system compared with the mode space multiplication by the cavity filters, in which a single feedback loop for a filtering cavity is required. However, the interleaver stabilization is robust and easy to implement compared with the stabilization of the high-finesse cavity, because the capture range of the interleaver stabilization covers half of the wavelength, and that of the cavity stabilization is the wavelength divided by the finesse.
It is interesting to discuss how high the OFC mode spacing can be multiplied by interleaving and the phase demodulation. In this study, the limit is determined by the bandwidth of the arbitrary function generator (200 MHz) for the phase demodulation. If a faster arbitrary function generator is used, the mode spacing of the OFC is expected to extend up to bandwidth of the WG-EOM (10 GHz) without power loss. In the case of a higher multiplication factor, the acceptable range to the path length error for the pulse envelope repetition rate multiplication becomes narrow as mentioned before, and if it is the order of the carrier wavelength, the phase shift induced by the interleaver is no longer adjustable without the deterioration of the pulse envelope repetition rate multiplication. As shown in Fig. 2(d), 100 $\mu$m of the interleaver path length error is allowed for the pulse envelope repetition rate at 400 MHz. Then, for the acceptable range of the interleaver path length error of 1550 nm, the highest pulse envelope repetition rate is expected to be $\sim 25$ GHz. If a faster WG-EOM is used (e.g., 40 GHz), the acceptable range of the interleaver path length error can determine the largest mode spacing available with the interleaving technique. For further multiplication of the mode spacing, the cavity filtering technique can be employed in series at the expense of the optical power. The total optical power loss is substantially reduced by the interleaving technique, compared with the mode spacing multiplication only with the filtering cavity.
4. Conclusions
We demonstrate multiplication of the OFC mode spacing by means of interleaving technique by a factor of 4. With this technique, the mode spacing multiplication can be carried out without power loss, which is in contrast to the mode filtering technique [5,6]. After interleaving, the optical pulse envelope repetition rate is multiplied, but the mode spacing of the OFC is not multiplied. The carrier phase modulation of the optical pulse train is responsible for the failure of the mode spacing multiplication. For the mode spacing multiplication of the OFC, the carrier phase is demodulated by the WG-EOM. As a result of the interleaving and the phase demodulation, the mode spacing of the OFC is indeed quadrupled. In addition, the optical paths in the interleavers are stabilized for the stable operation of the mode spacing multiplication. In the current study, half of the optical power is lost owing to the polarization alignment, but for lossless mode spacing multiplication, the fast polarization control technique [22] can be employed. Further multiplication (by 8 or 16) and optical bandwidth broadening of interleaved OFCs by a highly-nonlinear fiber are for future subjects.
Funding
Japan Society for the Promotion of Science (21K04930).
Disclosures
The authors declare 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.
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