Mode density multiplication of an optical frequency comb by N<sup>2</sup> with phase modulation

Taro Hasegawa

doi:10.1364/OE.26.024551

1. Introduction

An optical frequency comb (OFC) was originally developed for precise metrology as a frequency ruler in the optical region. The remarkable performance in the frequency measurements extends applications of the OFCs to various fields such as distance measurements [1], optical communications [2], and astronomy [3]. One of the successful applications is direct-comb spectroscopy, in which the OFCs are used as broadband light sources. In direct-comb spectroscopy, spectroscopic information such as absorption by molecular gas is recorded onto intensity of the individual mode of the OFC. To extract the information, several schemes such as dual-comb spectroscopy [4,5], two-dimensional spectroscopy [6], and continuous-wave-comb heterodyne (CCH) spectroscopy [7–10] have been employed.

Spectral resolution of direct-comb spectroscopy is limited by the mode separation of the OFC. For OFCs based on mode-locked lasers, the separation is equal to the pulse repetition rate. It is approximately confined between 20 MHz and 10 GHz because of practical length of the laser cavity. Increase of the mode separation is useful for some applications of the OFCs such as an astro-comb [3] and two-dimensional spectroscopy [6], in which precise and broadband but sparse frequency markers are required. To this end, the dense modes of the OFC are thinned out by an optical cavity [11, 12]. The OFCs with high repetition rate have also been developed using a small-size cavity [13] or fast modulation of continuous-wave (cw) lasers [14,15]. Combination of the small-size cavity and the modulation of cw lasers is also reported [16,17].

In the opposite direction, reduction of the mode separation is also useful to improve the spectral resolution of direct-comb spectroscopy and to increase information density in optical communications. However, few works of low-repetition-rate OFCs were reported because stable mode-lock operation is difficult for long laser cavities. Mode-locked lasers with a repetition rate less than 1 MHz were exceptionally realized using a special setup [18], but it has not been reported whether or not these mode-locked lasers have comb-like mode distribution in frequency domain. By using OFCs based on cw lasers with modulation, the mode spacing can be directly determined by the modulation frequency, and such OFCs are applied to spectroscopy [19]. However, broad spectral bandwidth of OFCs of mode-locked lasers is beneficial to spectroscopy [20], and therefore low-repetition-rate OFCs based on mode-locked lasers are useful for high-resolution broadband spectroscopy. High-resolution direct-comb spectroscopy has been carried out by sweeping the OFC mode frequencies [21–23], but for fast measurement without sweeping time, OFCs with high mode density are often preferred.

To decrease the repetition rate of the OFC (or equivalently to increase the mode density), the use of modulation techniques is another approach. When an output light of the OFC passes through an optical modulator, sidebands generated by the modulator fill the frequency gap between the original OFC modes. By applying a tailored sequence of the modulation signal to the modulator, the mode density of the modulated OFC is multiplied. A straightforward scheme to reduce the repetition rate of the OFC is the down sampling (intensity modulation) [24–26]. In this technique, however, the average optical power of the OFC becomes 1/P(P is the mode-density multiplication factor) after the modulation, and hence power amplification is required before or after the down sampling. In contrast, phase modulation has remarkable advantage over intensity modulation because no optical power is dissipated in principle. By the phase modulation, envelope structure of the pulse train does not modified. Thus the intensity repetition rate of the OFC is unchanged, but the mode density in frequency domain is certainly multiplied. Then the peak intensity in time domain with the phase modulation is smaller than that with the down sampling for identical optical average power. The peak power suppression of the phase-modulated OFC is worthwhile for some applications in which optical detector saturation should be avoided, such as dual-comb spectroscopy and ranging using the OFC [27]. Especially, application of the phase-modulated OFC to dual-comb spectroscopy is promising for broadband and high-resolution spectroscopy. The phase modulation applied to mode-locked lasers is utilized for attosecond optical pulse generation as well [28,29].

For various applications, every mode of the modulated comb is desired to have the same mode intensity. For this purpose, two phase modulation sequences based on the spectral Talbot effect [30] and on the pseudo-random binary sequence (PRBS) [31] were studied so far.

In this paper, we introduce another simple modulation sequence for the mode density multiplication of the OFC by P = N² (N is a positive integer). This sequence allows multiplication factors that are not accessible in the PRBS modulation (e.g., 4² = 16). In the previous work [30], the modulation signal is P-valued. In contrast, the modulation signal introduced in this paper is N-valued, where the multiplication factor P is N ². Thus the modulation signal amplitude does not have to be set so accurately as that in the previous work [30]. More importantly, the present sequence can be used together with those of [30] and [31]. We have demonstrated that the mode density of an Er-doped fiber laser OFC is multiplied by P = 4², 8², and 3 · 4². In the case of the multiplication by 3 · 4², the present phase modulation sequence is superimposed on the PRBS. Direct-comb spectroscopy (CCH spectroscopy of methane) has also been carried out using the mode-density-multiplied OFC by 4².

2. Principle

Suppose that the j-th pulse of the OFC is phase-modulated by adding a phase of ϕ_j. The modulation is periodic, as

ϕ_{j + P} = ϕ_{j},

where P is the number of pulses in one phase-modulation period which is identical to the factor of mode-density multiplication (MDM).

Optical electric field of the phase-modulated OFC light is expressed in time domain as,

E (t) = E_{0} \sum_{j = - \infty}^{\infty} g (t - j t_{r}) \exp [- i ω_{c} (t - j t_{r})] \exp [i (j ϕ_{o} + ϕ_{j})],

where E₀ is electric field amplitude, t_r is pulse repetition time (period) of the OFC, ω_c/2π is center frequency of the OFC, ϕ_o is carrier-envelope offset phase, and g(t) is the envelope of a single OFC pulse. By the Fourier transformation of Eq. (2), the electric field in frequency domain turns out to be

\begin{array}{l} \tilde{E} (ω) = \int_{- \infty}^{\infty} E (t) \exp (i ω t) d t \\ = E_{0} \frac{ω_{r}}{P} \tilde{g} (ω - ω_{c}) \sum_{k = - \infty}^{\infty} δ (ω + ω_{o} - k \frac{ω_{r}}{P}) {\hat{Φ}}_{(k \mod P)}, \end{array}

where ω_r/2π is the repetition frequency (ω_r = 2π/t_r), ω_o is the carrier-envelope offset angular frequency (ϕ_o/t_r), δ is the Dirac delta function,

\tilde{g} (ω)

is the Fourier-transformed function of g(t), and (k mod P) is remainder after division of k by P. Here,

{\hat{Φ}}_{k^{'}} (0 \leq k^{'} < P)

is the k′-th component of the discrete Fourier transformation (DFT) of

e^{i ϕ_{j}}

, namely,

{\hat{Φ}}_{k^{'}} = \sum_{j = 0}^{P - 1} \exp (i ϕ_{j}) \exp (2 π i \frac{j k^{'}}{P}) .

The expression of Eq. (3) implies that the spectral separation between the neighboring comb modes reduces to ω_r/(2πP) in frequency, and the mode amplitude (in complex expression) is given by ${\hat{Φ}}_{(k \mod P)}$ . The phase modulation generates sidebands, which fill the gap between the original OFC modes.

When the phase-modulated OFC is utilized for various applications, each OFC mode is desired to have equal intensity. For this purpose, the modulation sequence ϕ_j is tailored so that $| {\hat{Φ}}_{k^{'}} |$ is constant regardless of the index k′. In the previous works, the value of ϕ_j was taken as $\frac{q}{P} π j^{2}$ (q is an integer that is mutually prime to P) in [30], and as the PRBS in [31]. Here we introduce another phase sequence as

ϕ_{N j^{'} + j^{″}} = 2 π \frac{j^{'} j^{″}}{N},

where j′ and j″ are integers between 0 and N − 1, and j = Nj′ + j″. It is easily understood that ϕ_j is N-valued, and the period P is N². This phase sequence gives equal intensity to each mode as shown in Appendix A. For example, when N = 4 (P = 16), the phase sequence is,

ϕ_{j} = {\begin{array}{l} 0 & for j = 0, 1, 2, 3, 4, 8, 10, 12 \\ π / 2 & for j = 5, 15 \\ π & for j = 6, 9, 11, 14 \\ 3 π / 2 & for j = 7, 13 \end{array}

which is given in Fig. 1(a) in orange with the pulse envelope of the OFC in black.

Fig. 1 (a) Schematic diagram of pulse train in black and a phase-modulation sequence of Eq. (6) in orange. This phase modulation increases the mode density of the OFC by a factor of 4². (b) Pulses categorized into four groups according to the remainders of the pulse number divided by four. In the first group, the pulse numbers are the multiple of 4, and in the second (third, fourth) group, those are the multiple of 4 plus 1 (2,3). The carrier phase shift for the pulses is also given. (c) The mode spectra of the OFC without and with the phase modulation. The black, red, green, and blue modes correspond to the first-, second-, third-, and fourth-group pulses, respectively. (d) An alternative phase modulation sequence giving the same mode spectrum as (a).

Download Full Size | PDF

3. Physical insight of the mode-density multiplication

We propose the phase modulation sequence of Eq. (5) and Fig. 1(a) for the MDM by N = 4 (P = 16). At first glance, it is hard to understand that the modulated OFC has the desired properties, but a clear explanation is available as follows.

Here, we consider the specific case of N = 4 (P = 16). Sixteen OFC pulses in one period of the phase modulation are numbered from 0 to 15 in order [Fig. 1(a)], and categorized into 4 groups as in Fig. 1(b). The first group consists of the pulses whose number is a multiple of 4, namely, 0, 4, 8, and 12. The phase modulator provides the phase shift of 0 (no phase shift) to these pulses. For this group, the pulse interval time is 4t_r [t_r is the pulse repetition time of the original OFC as in Eq. (2)]. Therefore, in frequency domain, the pulses of the first group provide the OFC modes with spacing of (ω_r/2π)/4 [black modes in Fig. 1(c)] and with the same carrier-envelope offset frequency as the original OFC [ω_o/2π in Eq. (3)]. The second group consists of the pulses whose number is a multiple of 4 plus 1, namely, 1, 5, 9, and 13. The phase shifts applied to the pulses of numbers 1, 5, 9, and 13 are zero, π/2, π, and 3π/2, respectively. Hence the applied phase increases by π/2, and this phase increase is equivalent to the carrier-envelope offset phase shift. As a result, the pulses of the second group provide comb modes with the mode spacing of (ω_r/2π)/4 and the carrier-envelope offset frequency of (ω_o/2π) + (ω_r/2π)/16 [red modes in Fig. 1(c)]. Similarly, the third and fourth groups of the pulses, whose numbers are multiples of 4 plus 2 and 3, respectively, provide comb modes with the offset frequency of (ω_o/2π) + 3(ω_r/2π)/8 and (ω_o/2π + 3(ω_r/2π)/16, respectively [green and blue modes in Fig. 1(c)]. The entire mode spectrum of the modulated OFC is a simple sum of that of the four groups, since they have different frequency components, and no interference exists. The mode intensities are equal to each other.

The physical insight mentioned above suggests variations of the phase modulation sequence which also provides homogeneously-distributed multiplied optical modes. Specifically, the phase increases for the four groups are exchangeable, and adding constant phase to every pulse in one group does not affect the mode spectrum in intensity. Figure 1(d) gives an example, in which the phase shift increases by 0, π, π/2, and 3π/2, for the first, second, third, and fourth groups, respectively, and the constant phases of π/2, 3π/2, and π are added to the second, third, and fourth groups, respectively. This phase modulation sequence also provides the homogeneous distribution of the mode-density multiplied OFC.

4. Experiment and discussions

Optical spectrum of the phase-modulated OFC is observed by means of heterodyne detection using a single-mode cw laser as a local oscillator. Figure 2(a) shows experimental setup. The output from an OFC based on a home-made nonlinear-polarization-rotation mode-locked Er-doped fiber laser (ω_r/2π = 66.87 MHz) is amplified by a fiber amplifier, followed by a highly-nonlinear fiber to broaden the OFC spectrum. The repetition rate of the OFC is phase-locked to a radio-frequency (rf) oscillator. The output light is divided into two, and one is used for the cw laser frequency stabilization described later. The other light undergoes the phase modulation mentioned above in an electro-optical phase modulator (EOM, Thorlabs, LN53S-FC). The modulation signal of Eq. (5) is provided by an arbitrary function generator (NF Corporation, WF1968). The modulation signal amplitude is set in accordance with the half-wave voltage V_π of the EOM (8.9 V).

Fig. 2 (a) Experimental setup. A glass cell (GC) filled with methane is inserted for spectroscopy. (b) and (c) are beat-note spectral lines at 20 MHz recorded by the spectrum analyzer with RBW of 1 kHz and 1 Hz, respectively, when the ECLD frequency is stabilized with respect to one of the OFC modes. In (c), the linewidth is limited by RBW of the spectrum analyzer. Abbreviations: FA for an optical fiber amplifier, HNLF for a highly-nonlinear fiber, BS for a 50:50 beam splitter, AFG for an arbitrary function generator, FC for a fiber coupler, BPD for balanced photodetectors, SA for an rf spectrum analyzer, BPF for an rf bandpass filter at 20 MHz, RFA for an rf amplifier (15 dB), M for an rf mixer, PZT for a piezoelectric transducer to adjust the laser cavity length, and LO for a local oscillator.

Download Full Size | PDF

The phase-modulated light is spatially dispersed by a diffraction grating (600 grooves/mm), and a spectral component within (1645 ± 1) nm is then overlapped with a single-mode cw laser light at 1645 nm at a two-port fiber-based 50:50 beam splitter (Thorlabs, TW1550R5A2). The cw laser is a home-made external-cavity laser diode (ECLD). The overlapped light is detected by balanced photodetectors (Thorlabs, PDB480C-AC), and the detected beat notes are resolved by an rf spectrum analyzer (Rohde Schwarz, FSC3). The balanced detection eliminates irrelevant rf spectral components at the repetition rate of the original OFC and its harmonics, and only the beat-note spectrum is observed [10].

The cw laser frequency is stabilized to the nearest comb mode by the optical phase-locking technique [32] with the offset frequency of 20 MHz, which is provided by another rf oscillator (local oscillator). The error signal is fed back to a driving current of the ECLD and to a piezoelectric transducer for adjustment of the ECLD cavity length. The two rf oscillators, the arbitrary function generator, and the rf spectrum analyzer are electrically synchronized to each other. Figure 2(b) shows rf spectrum of the beat note between the OFC and the phase-locked ECLD. The resolution bandwidth (RBW) of the rf spectrum analyzer is 1 kHz. From this result, the servo loop bandwidth turns out to be about 160 kHz. With smaller RBW of 1 Hz, the spectrum is shown in Fig. 2(c), in which the spectral linewidth is determined by the RBW. This result implies that the relative frequency fluctuation is less than 1 Hz.

Figure 3 shows the observed rf spectra of the beat notes. Figure 3(a) is the result with no phase modulation, Fig. 3(b) is with phase modulation of N = 4 (P = 16), Fig. 3(c) shows enlarged spectrum of Fig. 3(b), Fig. 3(d) is with phase modulation of N = 8 (P = 64), and Fig. 3(e) shows enlarged spectrum of Fig. 3(d). Figures 3(b) and 3(c) [Figs. 3(d) and 3(e)] show homogeneous mode intensity distribution, while the mode density is multiplied by factors of 16 (64).

Fig. 3 Spectra of the beat notes of the ECLD with (a) the original OFC, and with the phase-modulated OFC for (b) P = 16, (d), 64, and (f) 48, respectively. The RBW of the spectrum analyzer is 300 kHz. The repetition frequency of the original OFC is indicated by dotted lines. In order to distinguish each OFC mode clearly, figures of (c), (e), and (g) are enlarged spectra of (b), (d), and (f), respectively and recorded with the RBW of 10 kHz.

Download Full Size | PDF

Here, the maximum value of the multiplication factor is discussed. Figure 2(b) shows that the beat note between the original OFC and the ECLD has a signal-to-noise ratio of about 58 dB for RBW of 1 Hz. The noise level is determined by the shot noise of the total optical power into the balanced photodetectors, as in [10]. In comparison, Fig. 3 illustrates that the mode density multiplication by the phase modulation reduces the individual intensity of the beat notes to 1/P. On the other hand, the noise intensity remains constant, because it is determined by the shot noise level for the total optical power level incident on the detectors, which is unchanged by the phase modulation. Suppose that the signal-to-noise ratio of 10 dB is required for a certain practical application. Then the MDM factor is able to be increased up to 10^(58−10)/10 ~ 63 000 ~ 250² for 1 s data accumulation in the present setup. In such a case, the phase modulation signal applied to the EOM is 250-valued and is required to be set as accurately as 2V_π/250, which can be provided by 8-bit arbitrary function generators. The required phase accuracy is considerably lower than that required in [30] for the identical multiplication factor. It is also worth noting here that the relative frequency stability of the phase-modulated OFC and the ECLD is essential to resolve the beat notes in the case of high MDM factor. The mode separation of the modulated OFC is 1 kHz for the MDM factor of 63 000. Even in such a case, the linewidth in Fig. 1(c) is small enough to distinguish every OFC mode.

It is important to mention that the phase sequence for MDM can be superimposed on other phase modulation sequence such as [30,31]. Here we consider the case in which the OFC light undergoes two phase modulations in series: one is for the MDM by P₁, and the other is that by P₂, as shown in Fig. 4(a). Here, each phase modulation sequence is for homogeneously-distributed mode intensity such as sequences based on the spectral Talbot effect [30], PRBS [31], or that of Eq. (5). If P₁ and P₂ are prime to each other, the resultant optical modes of the OFC also have homogeneously-distributed mode intensity, and the mode density is multiplied by P₁ × P₂, as shown in Appendix B. We have demonstrated the case of the combination of the PRBS (P₁ = 3) and the sequence of Eq. (5) (P₂ = 16), resulting the total MDM factor of 48. Instead of the use of two EOMs as in Fig. 4(a), we use a single phase modulator, and the arbitrary function generator is set to produce the added phase sequence, as shown in Fig. 4(b). The result is shown in Fig. 3(f) [Fig. 3(g) is the enlarged spectrum], in which the MDM by 48 has been successfully carried out. The serial combination of two phase modulators is naturally extended to that of three or more modulators.

Fig. 4 Phase modulation for the mode-density multiplication of the OFC by factor of P₁ × P₂, where P₁ and P₂ are prime to each other. In (a), two electro-optic phase modulators (EOM1 and EOM2) are used in series. In (b), two modulation signals are added and applied to one EOM. AFG1 and AFG2 provide modulation signals for P₁ and P₂, respectively.

Download Full Size | PDF

5. Application to spectroscopy

The phase-modulated OFC for P = 16 is applied to spectroscopy of methane. Application to dual-comb spectroscopy is intriguing, as it becomes not only broadband but also high-resolution. As a first step, however, we applied the mode-density-multiplied OFC to CCH spectroscopy, which can be easily carried out with essentially the identical setup in Fig. 2(a) except inserting a 20 cm-long glass cell filled with methane at a pressure of 200 Pa between the EOM and the diffraction grating. The measurement procedure was described in [10]. Briefly, the ECLD wavelength is set at 1645.568 nm, which is close to the R(6) transitions of the 2ν₃ band of ¹²CH₄. Each beat note spectral line intensity in Fig. 3 contains the absorption information. The rf frequency range is limited from −2 GHz to 2 GHz by the nominal rf bandwidth of the balanced photodetectors of 1.6 GHz. Each beat note line intensity is measured by the rf spectrum analyzer with the resolution bandwidth of 300 kHz and averaged over 100 measurements, so that the measurement time for one point is 0.3 ms.

The observed intensities of the beat notes mainly depend on the rf response of the balanced photodetectors and the following electronics [10]. To extract absorption spectrum of methane, they are normalized by those recorded when the ECLD wavelength is set at 1645.210 nm, where there are no absorption lines of methane in the vicinity. Black dots in Figs. 5(a) and 5(b) show the normalized spectrum recorded using the original OFC and the phase-modulated OFC for P = 16, respectively. In Fig. 5(b), the spectral resolution is 4.18 MHz, which is 1/16 of the resolution with the original OFC (66.87 MHz).

Fig. 5 Absorption spectrum of the R(6) transitions of the 2ν₃ band of methane at 1645.57 nm recorded by cw-comb heterodyne spectroscopy using (a) the OFC without the modulation and (b) the phase-modulated OFC for P = 16. The horizontal axis indicates the optical frequency of the phase-modulated OFC relative to the cw laser frequency. The blue dotted lines indicate the center frequencies and the intensity of the absorption lines. The red curve is the fitted spectrum assuming the Doppler broadening at 293 K. (c) and (d) show the residuals of the normalized and fitted spectra in (a) and (b), respectively.

Download Full Size | PDF

The normalized spectra are fitted to a sum of three Doppler-broadened absorption lines at temperature of 293 K with fixed values of the Doppler linewidths, the relative line intensities, and the relative center frequencies of the three transitions, according to the HITRAN 2012 database [33]. The fitting parameters are the entire transition intensity, the background level, and the background slope. The fitted spectra are given in red curves in Figs. 5(a) and 5(b). The spectral line frequencies and the relative intensities are expressed by blue dashed lines together with the assignment, that is, the irreducible representation of the tetrahedral point group for the lower level of the transition. Figures 5(c) and 5(d) depict residuals between the normalized and fitted spectra in Figs. 5(a) and 5(b), respectively.

In Fig. 5(b), the absorption spectral lines are observed with 16 times more data points than those in Fig. 5(a), but some data points apparently deviate from the fitted curve. These points appear at the frequencies corresponding to the beat notes between the original OFC and the ECLD. The deviation is caused by polarization drift of the optical field incident to the EOM, in which the modulation depth depends on the polarization. The modulation depth for a certain polarization component (extraordinary ray in LiNbO₃ crystal in the EOM) is three times larger than that for the other polarization component (ordinary ray). If the phase modulation is optimized for the extraordinary ray, the sideband intensity is different from the original-OFC-mode intensity for the ordinary ray. Therefore, the polarization drift may cause the intensity drift of the original OFC modes. Temperature change and some mechanical effect induce the drift of the OFC polarization and the polarization transmission in a single-mode optical fiber linking the EOM with the OFC. We have to record signal and background spectra for normalization, and the polarization condition into the EOM for the background measurement is slightly different from that for the signal measurement. To reduce standard deviations of the measurement, it is the easiest way to eliminate the data points of the original OFC modes. Indeed, the standard deviation reduces down to 43 % by the data elimination. We expect that the polarization drift is diminished by the use of polarization-maintaining fibers. Another scheme to avoid this deviation will be published in a separate paper.

In the previous section, we pointed out that the phase modulation of the OFC output reduces the signal-to-noise ratio of the beat-note spectrum to 1/P. In fact, the standard deviation of the residual spectrum in Fig. 5(d) without the data points at the original OFC modes is 1.22 times larger than that without MDM in Fig. 5(c). The deterioration in the signal-to-noise ratio is an inevitable expense for higher spectral resolution, which is important to observe narrow-linewidth spectral lines for nonlinear spectroscopy and detailed spectral line shape for optical thermometry.

The signal-to-noise ratio is improved for longer averaging time. For the result in Fig. 5, the averaging time for one data point measurement is 0.3 ms. However, total measurement time to observe the entire spectral line in Fig. 5 is as long as 5 minutes, because of the use of the rf spectrum analyzer, with which many irrelevant measurements are carried out. In order to shorten the entire measurement time, the rf spectrum analyzer used here should be replaced with a fast digitizer and a computer. With this replacement, longer averaging time becomes practical, and better signal-to-noise ratio can be expected. For this purpose, the frequency stabilization of the ECLD is again crucially important.

6. Conclusions

The mode density of the OFC has been multiplied by the novel sequence of phase modulation. In the present scheme, the N-valued phase modulation gives the multiplication factor of N². The MDM factor can be chosen more arbitrarily than in the PRBS. Especially for Fourier-transform spectroscopy, the MDM by 2^m, which is not available with the PRBS, is preferred for fast Fourier transformation. Physical insight for the homogeneous mode distribution is given. In the demonstration, the mode density of an Er-doped fiber laser OFC has been multiplied by 16, 64, and 48, and successfully confirmed by the beat-note measurement with a cw laser. Particularly, the case of multiplication factor of 48 has been carried out by combining the present scheme with the PRBS.

Application of the MDM by 16 to absorption spectroscopy of methane at 1645 nm has been successfully demonstrated. At present, the data acquisition is carried out by the rf spectrum analyzer, but time-domain measurement by the use of a fast digitizer should be introduced for long data accumulation time to improve the signal-to-noise ratio of CCH spectroscopy and for a higher MDM factor. Dual-comb spectroscopy with the phase-modulated OFC is for future subject.

A. DFT of the phase sequence of Eq. (5)

The DFT of the phase sequence of Eq. (5) is

{\hat{Φ}}_{k} = \sum_{j^{'} = 0}^{N - 1} \sum_{j^{″} = 0}^{N - 1} \exp (2 π i \frac{j^{'} j^{″}}{N}) \exp [2 π i \frac{(N j^{″} + j^{'}) k}{N^{2}}] .

Equation (7) is reformed as,

{\hat{Φ}}_{k} = \sum_{j^{'} = 0}^{N - 1} \sum_{j^{″} = 0}^{N - 1} \exp [2 π i \frac{j^{″} (j^{'} + k)}{N}] \exp (2 π i \frac{j^{'} k}{N^{2}}) .

Using the orthogonality of complex exponentials, namely,

\sum_{j^{″} = 0}^{N - 1} \exp (2 π i \frac{j^{″} k^{'}}{N}) = N δ_{k^{'} \mod N, 0},

where δ_a,b is the Kronecker’s delta, the sum about j″ in Eq. (8) is simplified. Then we obtain

\begin{array}{l} {\hat{Φ}}_{k} = \sum_{j^{'} = 0}^{N - 1} N \exp (2 π i \frac{j^{'} k}{N^{2}}) δ_{(j^{'} + k) \mod N, 0} \\ = N \exp (2 π i \frac{(N - k) k}{N^{2}}) . \end{array}

Equation (10) implies that the absolute value of ${\hat{Φ}}_{k}$ is N for any value of k.

B. Mode distribution with two serial phase modulations

Here we suppose that the OFC light undergoes two phase modulations of $e^{i ϕ_{j}^{(1)}}$ and $e^{i ϕ_{j}^{(2)}}$ . The phases $ϕ_{j}^{(1)}$ and $ϕ_{j}^{(2)}$ are periodic, namely,

ϕ_{j + P_{n}}^{(n)} = ϕ_{j}^{(n)},

where n is 1 or 2, and it is supposed that periods of P₁ and P₂ are prime to each other.

The DFT of $e^{i ϕ_{j}^{(1)}}$ and $e^{i ϕ_{j}^{(2)}}$ is,

{\hat{Φ}}_{k_{n}}^{(n)} = \sum_{j = 0}^{P_{n}} \exp [i ϕ_{j}^{(n)}] \exp (2 π i \frac{j k_{n}}{P_{n}}) .

For homogeneously-distributed OFC modes, $| {\hat{Φ}}_{k_{1}}^{(1)} |$ does not depend on k₁, and similarly, $| {\hat{Φ}}_{k_{2}}^{(2)} |$ does not depend on k₂.

By the inverse DFT, we have

\exp [i ϕ_{j}^{(n)}] = \frac{1}{P_{n}} \sum_{k_{n} = 0}^{P_{n}} {\hat{Φ}}_{k_{n}}^{(n)} \exp [- 2 π i \frac{j k_{n}}{P_{n}}] .

Here, it should be noted that in Eq. (13) the value of j does not have to be between 0 and P_n − 1.

As a result of the two phase modulations in series, the OFC mode distribution is expressed by the DFT of $e^{i ϕ_{j}^{(1)}} \times e^{i ϕ_{j}^{(2)}}$ , which is, noting that the total phase sequence is P₁P₂-periodic,

{\hat{Φ}}_{k} \sum_{j = 0}^{P_{1} P_{2} - 1} \exp [i ϕ_{j}^{(1)}] \exp [i ϕ_{j}^{(2)}] \exp (2 π i \frac{j k}{P_{1} P_{2}}) .

By substituting Eq. (13) into Eq. (14), we obtain,

{\hat{Φ}}_{k} = \frac{1}{P_{1} P_{2}} \sum_{j = 0}^{P_{1} P_{2} - 1} \sum_{k_{1} = 0}^{P_{1} - 1} \sum_{k_{2} = 0}^{P_{2} - 1} {\hat{Φ}}_{k_{1}}^{(1)} {\hat{Φ}}_{k_{2}}^{(2)} \exp (- 2 π i j \frac{P_{2} k_{1} + P_{1} k_{2}}{P_{1} P_{2}}) \exp (2 π i \frac{j k}{P_{1} P_{2}}),

where k is an integer between 0 and P₁P₂ − 1.

Summation about j can be carried out with the use of the orthogonality of complex exponents, which is expressed as,

\sum_{j = 0}^{P_{1} P_{2} - 1} \exp (2 π i j \frac{P_{2} k_{1} + p_{1} k_{2}}{P_{1} P_{2}}) \exp (2 π i \frac{j k}{P_{1} P_{2}}) = {\begin{array}{l} P_{1} P_{2} & for (P_{2} k_{1} + P_{1} k_{2}) \mod P_{1} P_{2} = k \\ 0 & otherwise \end{array} .

Equation (16) implies that only one term of the right hand side of Eq. (15) is non-zero for given value of k, and its proof is as follows. Suppose that k₁ and k₂ satisfy the relation

(P_{2} k_{1} + P_{1} k_{2}) \mod P_{1} P_{2} = k

for given value of k, and that

k_{1}^{'}

and

k_{2}^{'}

also satisfy

(P_{2} k_{1}^{'} + P_{1} k_{2}^{'}) \mod P_{1} P_{2} = k .

Here, k_n and $k_{n}^{'}$ are integers between 0 and P_n − 1. Equations (17) and (18) can be rewritten as,

P_{2} k_{1} + P_{1} k_{2} - k = a P_{1} P_{2}

P_{2} k_{1}^{'} + P_{1} k_{2}^{'} - k = b P_{1} P_{2},

where a and b are integers. By subtracting Eq. (20) from Eq. (19),

P_{2} (k_{1} - k_{1}^{'}) + P_{1} [k_{2} - k_{2}^{'} + (b - a) P_{2}] = 0 .

Because P₁ and P₂ are prime to each other, the value of $k_{1} - k_{1}^{'}$ must be multiple of P₁ to satisfy Eq. (21). This condition is fulfilled only when $k_{1} - k_{1}^{'}$ , taking the definition that k₁ and $k_{1}^{'}$ are between 0 and P₁ − 1 into account. Similarly, it is obtained that $k_{2} = k_{2}^{'}$ . This result means that only one term of the right hand side of Eq. (15) is non-zero for given value of k. Noting that the right hand side of Eq. (15) consists of P₁P₂ terms and that 0 ≤ k < P₁P₂, a value of k corresponds to a pair of k₁ and k₂ one-to-one (in mathematical term, mapping from k into a pair of k₁ and k₂ is bijective). Therefore, Eq. (15) is expressed as,

{\hat{Φ}}_{k} = {\hat{Φ}}_{k_{1}}^{(1)} {\hat{Φ}}_{k_{2}}^{(2)},

where k₁ and k₂ satisfy the relation of Eq. (17). Then

| {\hat{Φ}}_{k} |

does not depend on k. From the analysis given in Appendix A, it can be concluded that the optical modes of the OFC with the two phase modulations in series have homogeneously-distributed intensity, and its density is multiplied by P₁P₂, if P₁ and P₂ are prime to each other.

Funding

Japan Science and Technology Agency (JST) through the ERATO MINOSHIMA Intelligent Optical Synthesizer (IOS) Project and JSPS KAKENHI Grant Number 15K05547.

Acknowledgments

The author acknowledges H. Sasada for discussions and comments.

References and links

1. P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17, 9300–9313 (2009). [CrossRef] [PubMed]

2. J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8, 375–380 (2014). [CrossRef] [PubMed]

3. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321, 1335–1337 (2008). [CrossRef] [PubMed]

4. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. 29, 1542–1544 (2004). [CrossRef] [PubMed]

5. I. Coddington, W. C. Swann, and N. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008). [CrossRef] [PubMed]

6. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445, 627–630 (2007). [CrossRef] [PubMed]

7. K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and single-wavelength laser,” Appl. Phys. Lett. 101, 051105 (2012). [CrossRef]

8. J-D. Deschênes and J. Genest, “Frequency-noise removal and on-line calibration for accurate frequency comb interference spectroscopy of acetylene,” Appl. Opt. 53, 731–735 (2014). [CrossRef] [PubMed]

9. D. A. Long, A. J. Fleisher, D. F. Plusquellic, and J. T. Hodges, “Multiplexed sub-Doppler spectroscopy with an optical frequency comb,” Phys. Rev. A 94, 061801 (2016). [CrossRef]

10. T. Hasegawa and H. Sasada, “Direct-comb molecular spectroscopy by heterodyne detection with continuous-wave laser for high sensitivity,” Opt. Express 25, A680–A688 (2017). [CrossRef] [PubMed]

11. M. S. Kirchner, D. A. Braje, T. M. Fortier, A. M. Weiner, L. Hollberg, and S. A. Diddams, “Generation of 20 GHz, sub-40 fs pulses at 960 nm via repetition-rate multiplication,” Opt. Lett. 34, 872–874 (2009). [CrossRef] [PubMed]

12. T. Hasegawa, “Broadened optical spectrum of repetition-rate-multiplied erbium-doped fiber comb,” J. Opt. Soc. Am. B 33, 2057–2061 (2016). [CrossRef]

13. A. Bartels, D. Heinecke, and S. A. Diddams, “Passively mode-locked 10 GHz femtosecond Ti:sapphire laser,” Opt. Lett. 33, 1905–1907 (2008). [CrossRef] [PubMed]

14. C.-B. Huang, Z. Jiang, D.E. Leaird, and A.M. Weiner, “High-rate femtosecond pulse generation via line-by-line processing of phase-modulated CW laser frequency comb,” Electron. Lett. 19, 1114–1115 (2006). [CrossRef]

15. X. Yan, X. Zou, W. Pan, L. Yan, and J. Azaña, “Fully digital programmable optical frequency comb generation and application,” Opt. Lett. 43, 283–286 (2018). [CrossRef] [PubMed]

16. P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012). [CrossRef]

17. C. Bao, P. Liao, A. Kordts, M. Karpov, M. Pfeiffer, L. Zhang, Y. Cao, G. Xie, C. Liu, Y. Yan, A. Almaiman, A. Mohajerin-Ariaei, A. Fallahpour, M. Tur, T. Kippenberg, and A. Willner, “Tunable insertion of multiple lines into a Kerr frequency comb using electro-optical modulators,” Opt. Lett. 42, 3765–3768 (2017). [CrossRef] [PubMed]

18. S. Kobtsev, S. Kukarin, and Y. Fedotov, “Ultra-low repetition rate mode-locked fiber laser with high-energy pulses,” Opt. Express 16, 21936–21941 (2008). [CrossRef] [PubMed]

19. G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10, 27–30 (2016). [CrossRef]

20. S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 µ m,” Appl. Phys. Express 8, 082402 (2015). [CrossRef]

21. T. Shioda, K. Fujii, K. Kashiwagi, and T. Kurokawa, “High-resolution spectroscopy using interleaved 100 GHz optical frequency comb scanned by phase modulator,” Opt. Commun. 284, 5180–5184 (2011). [CrossRef]

22. A. Nishiyama, S. Yoshida, Y. Nakajima, H. Sasada, K. Nakagawa, A. Onae, and K. Minoshima, “Doppler-free dual-comb spectroscopy of Rb using optical-optical double resonance technique,” Opt. Express 24, 25894–25904 (2016). [CrossRef] [PubMed]

23. S. Meek, A. Hipke, G. Guelachvili, T. Hänsch, and N. Picqué, “Doppler-free Fourier transform spectroscopy,” Opt. Lett. 43, 162–165 (2018). [CrossRef] [PubMed]

24. D. Mandridis, I. Ozdur, F. Quinlan, M. Akbulut, J. Plant, P. Juodawlkis, and P. Delfyett, “Low-noise, low repetition rate, semiconductor-based mode-locked laser source suitable for high bandwidth photonic analog-digital conversion,” Appl. Opt. 49, 2850–2857 (2010). [CrossRef] [PubMed]

25. A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, S. Aozasa, A. Mori, H. Nakano, A. Takada, and M. Koga, “Octave-spanning frequency comb generated by 250 fs pulse train emitted from 25 GHz externally phase-modulated laser diode for carrier-envelope-offset-locking,” Electron. Lett. 46, 1343–1344 (2010). [CrossRef]

26. D. Cole, S. Papp, and S. Diddams, “Downsampling of optical frequency combs,” J. Opt. Soc. Am. B 35, 1666–1673 (2018). [CrossRef]

27. X. Zhao, X. Qu, F. Zhang, Y. Zhao, and G. Tang, “Absolute distance measurement by multi-heterodyne interferometry using an electro-optic triple comb,” Opt. Lett. 43, 807–810 (2018). [CrossRef] [PubMed]

28. A. Baltuška, M. Uiberacker, E. Goulielmakis, R. Kienberger, V. S. Yakovlev, T. Udem, T. W. Hänsch, and F. Krausz, “Phase-controlled amplification of few-cycle laser pulses,” IEEE J. Sel. Top. Quantum Electron. 9, 972–989 (2003). [CrossRef]

29. C. Gohle, J. Rauschenberger, T. Fuji, T. Udem, A. Apolonski, F. Krausz, and T. Hänsch, “Carrier envelope phase noise in stabilized amplifier systems,” Opt. Lett. 30, 2487–2489 (2005). [CrossRef] [PubMed]

30. A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21, 4139–4144 (2013). [CrossRef] [PubMed]

31. N. B. Hébert, V. Michaud-Belleau, S. Magnan-Saucier, J-D. Deschênes, and J. Genest, “Dual-comb spectroscopy with a phase-modulated probe comb for sub-MHz spectral sampling,” Opt. Lett. 41, 2282–2285 (2016). [CrossRef] [PubMed]

32. M. Ohtsu, “Frequency stabilization in semiconductor lasers,” Opt. Quantum Electron. 20, 283–300 (1988). [CrossRef]

33. L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013). [CrossRef]

Mode density multiplication of an optical frequency comb by N² with phase modulation

Abstract

1. Introduction

2. Principle

3. Physical insight of the mode-density multiplication

4. Experiment and discussions

5. Application to spectroscopy

6. Conclusions

A. DFT of the phase sequence of Eq. (5)

B. Mode distribution with two serial phase modulations

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (22)

Optics Express