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Abstract
We introduce and experimentally apply “cosine similarity” as an index for quantitatively evaluating the degree of change in the spectra of optical frequency combs. The cosine similarity with the original spectrum increased or decreased as the amount of control applied to the combs increased or decreased; this is considered to be an appropriate indication of spectral similarity. Therefore, we apply this approach to an evaluation of the temporal spectral changes in polarization-maintaining (PM) and non-PM combs. The results suggest that there is no significant difference between the spectral stabilities of PM and non-PM combs, and reveal that the spectral sensitivity to the amount of control is a more effective factor.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An optical frequency comb has a comb-like spectral structure with narrow modes equally spaced in the frequency domain and aligned over a wide bandwidth [1,2]. The broad and precise nature of the comb makes it possible to realize optical-microwave frequency links by self-referencing and beat detection with continuous-wave lasers at various wavelengths and is one of the main reasons why such combs are useful and powerful tools.
The usage range of the comb has expanded, and applications that utilize the entire broad spectrum of the comb, such as dual-comb spectroscopy [3–14], have developed greatly. In direct frequency comb spectroscopy including dual-comb spectroscopy, although the precise frequency of each comb mode is an important feature, it is also important that the comb spectrum itself is stable to allow us to obtain a precise transmission/absorption spectrum. In particular, if the spectrum is not sufficiently stable during the acquisition of the comb spectrum with and without the sample, the precision of the transmittance will be impaired. Some results suggesting the precision of dual-comb spectroscopy have been reported [15,16], and an evaluation of spectral stability is essential if we are to obtain more precise results. The spectral stability of continuous wave lasers and microwave oscillators has been evaluated [17–19], but a quantitative evaluation of broadband spectra such as optical frequency combs has yet to be reported. In this study, we introduce cosine similarity as a measure with which to evaluate the spectral stability of the comb. This is the first proposal of a quantitative method for evaluating the spectral stability of the comb. The cosine similarity is defined as the cosine of the plane angle between two vectors, which takes a value of 1 if the vectors have an identical orientation and 0 if they are orthogonal. Recently, this metric has been applied to pattern recognition and medical diagnosis [20,21]. We also evaluate the validity of the cosine similarity by experimentally measuring the spectral stability of the comb with different amounts of control for several actuators such as pump laser power and laser cavity length. In addition, we compare the spectral stability of combs composed of polarization-maintaining (PM) and non-PM fibers under a free-running condition and feedback control and discuss the direction we must take to obtain combs with high spectral stability.
2. Method
Cosine similarity is often used to evaluate the similarity between two multidimensional quantities. When a quantity is expressed as a multidimensional vector, if the two vectors have the same orientation, the plane angle θ between them is zero. Thus, cos θ is 1 when the two vectors are in the same direction, and 0 when they are orthogonal.
In this study, discrete spectral intensity distributions in the wavelength (or frequency) domain are regarded as multidimensional vectors, as shown in Fig. 1. The cosine similarity is the inner product normalized by the norms of the vectors and can be expressed as follows,
The similarity between two spectra can be quantitatively evaluated using the cosine similarity. In addition, in measurements using the ratio between spectra measured with and without samples such as transmittance measurements, the line shape of the samples is not distorted even if the spectral profiles are similar figures, and we assume that the cosine similarity, which is 1 even if the spectral profiles are similar figures, is also appropriate in this sense.
It is possible to evaluate not only how the spectrum changes under various conditions but also how the spectrum changes over time. For example, for spectra acquired continuously at a certain time interval, if we consider the first spectrum as the reference spectrum, the cosine similarities of the second and subsequent spectra show how they are changing from the first spectrum. In this paper, we quantitatively evaluate the temporal spectral stability of the comb using this scheme.
3. Experimental setup
The experimental setup is shown in Fig. 2. We used four custom-made optical frequency combs as the comb source with broad spectra ranging from 1400 nm to 1700 nm. For each comb, we divided the oscillator output into two branches, detected the fCEO and frep signals in one branch, and phase locked them to a reference signal as necessary to obtain a phase-stabilized comb. In the phase lock of the fCEO, the control signal was fed back to the injection current into the pump laser of the mode-locked fiber laser (oscillator), and in the phase lock of the frep, the control signal was fed back to the cavity length of the oscillator (controlled by PZT, EOM, and temperature control). In addition, the laser cavity is a fiber ring resonator, most of which is coiled on a thin copper plate. On the underside of the copper plate is a 40 mm square Peltier element, which mainly regulates the temperature of the single-mode fiber directly above the element. The approximate lengths of the laser cavity and the fiber directly above the element are shown in Table 1.
Fig. 2. Experimental setup. MLFL, mode locked fiber laser; EDFA, erbium-doped fiber amplifier; HNLF, highly nonlinear fiber; OSA, optical spectrum analyzer; P, polarizer; PZT, piezoelectric transducer; EOM, electro-optic modulator; TEC, temperature controller
Table 1. Main specifications of the comb systems. NALM: nonlinear amplifying loop mirror; NPR: nonlinear polarization rotation.
The comb output was assumed to be used in dual-comb spectroscopy, and the linearly polarized component was extracted with a polarizer, and then the spectrum was observed with an optical spectrum analyzer (OSA, Yokokawa 6375 or Ando 6315). Two of the four comb systems consisted of non-PM optical fibers, similar to those used in our previous studies (Comb #1 [22], Comb #2 [23]). The other two systems were newly fabricated comb systems consisting of PM optical fibers (PM Comb #1, PM Comb #2). The oscillators used as the comb sources for Comb #1, Comb #2, and PM Comb #1 utilized nonlinear polarization rotation (NPR) [24,25] as their mode-locking mechanism. In PM Comb #1, the linear polarization component was extracted by using a PBS at the oscillator output, and all the subsequent stages consisted of PM optical fibers. In PM Comb #2, all the fibers including the oscillator consisted of PM optical fibers, and a nonlinear amplifying loop mirror (NALM) was used for mode locking [26–28]. The comb system specifications are summarized in Table 1.
4. Results
In this section, we evaluate the spectral stability of the combs by acquiring their spectra with the experimental setup shown in Fig. 2, and evaluating these spectra using the cosine similarity described in Method. Figure 3 shows examples of the spectra output from the combs. In section 4.1, we evaluated how the spectrum of the comb changes by changing the amount of control applied to the actuators of the combs, using cosine similarity. This is also a consideration of whether cosine similarity is a valid metric for evaluating spectral similarity. In section 4.2, by using cosine similarity, we evaluate the temporal spectral stability of the combs under free-running and phase-stabilized conditions.
Fig. 3. Examples of the spectra output from the combs. The wavelength resolution of the spectra is 1 nm. S, power spectral density; Smax, Maximum value of power spectral density
4.1 Spectral changes when changing the control parameters of the frequency combs
We investigated how the cosine similarity changes when we increased or decreased the amounts of control applied to the actuators that control the frequency of the comb. The amounts of control are specifically the pumping power to the oscillator, the pumping power to the EDF, the laser cavity temperature of the oscillator, and the voltage applied to the EOM or PZT to control the cavity length.
Figure 4 shows the cosine similarity variation of the spectra with respect to the reference spectra for four combs when the amount of control was varied for only one actuator of interest. Here, the reference spectrum is that at the center value of the predetermined control amounts. The frep variations are observed with a frequency counter; the frep and fCEO were not phase locked in these measurements. We use the relative change Δfrep / frep for the amount of actuator control, since we consider frep to be the real control target for controlling the comb. The change in cosine similarity with respect to the amount of control appears to be close to a quadratic function. Therefore, we assumed that the cosine similarity y when the amount of control is changed by x (= Δfrep / frep) from zero can be expressed by the following equation,
By fitting Eq. (2) to the three points centered on Δfrep / frep = 0 in the results shown in Fig. 4, we calculated the coefficient β, which represents the degree of spectral change relative to the amount of control. The result is shown in Table 2. We consider that the cosine similarity appropriately evaluates the spectral similarity since the cosine similarity to the original spectrum decreases as the amount of control increases for all combs, and the change in the cosine similarity of each comb is consistent with the change in the sensory similarity of the spectrum.
Fig. 4. Cosine-similarity variation of comb spectra when the control amount applied to each actuator is changed. PZT, piezoelectric transducer; EOM, electro-optic modulator; Δfrep, amount of change in repetition rate due to control; frep, repetition rate; ΔP, amount of change in pump-laser power for erbium-doped fiber amplifier due to control; P, original laser power for the amplifier.
Table 2. The degree of spectral change β for the control amount for each comb. The values in parentheses are fitting errors.
The results described in this section suggest that cosine similarity is an appropriate index with which to evaluate the spectral similarity of optical frequency combs. Furthermore, from the results obtained using cosine similarity, we found that the degree of spectral change in response to the amount of control varied greatly among the combs (up to approximately 280 times among the four comb systems).
4.2 Temporal spectral stability evaluation of the optical frequency comb
In this section, we use cosine similarity to evaluate the temporal stability of the comb spectra with and without feedback control to stabilize frep and fCEO. Using a spectrum obtained at a certain time as a reference, we calculated the cosine similarity of spectra subsequent to the reference (see Method). Here we used an optical spectrum analyzer to acquire the spectrum of the comb intermittently every 100 seconds for 12 000 seconds. The measurement wavelength range was from 1400 nm to 1700 nm, and the number of measured wavelength points in the measurement at each time was 6001 (Yokokawa 6375) when measuring a free-running comb and 2001 (Ando 6315) when measuring a phase-stabilized comb. Here, we separately confirmed that there is little difference in cosine similarity even if the number of data points and OSA are different.
Figure 5 shows the spectral stability of the comb under a free-running condition ((a) and (b)). The laboratory was air-conditioned, and the temperature and humidity fluctuations were less than 0.3 °C and 3%, respectively, during the measurement, but the repeatability of the cosine similarity was low. The reason was not clear from the temperature and humidity, which were observed simultaneously. No significant difference in spectral stability was observed between the PM comb (a) and the non-PM comb (b).
Fig. 5. Examples of the spectral stability of the combs, time variation of the cosine similarities of the comb spectra with the reference spectrum at a certain time.
On the other hand, with the spectral changes (c) and (d) when frep and fCEO were stabilized by feedback control, the cosine similarity often decreased greatly compared with the free-running case. This was especially significant for a comb where the degree of spectral change was large relative to the amount of control (Comb #1 and PM Comb #1; see Table 2 in section 4.1). There was also no significant difference between the spectral stabilities of the non-PM (c) and PM (d) combs.
5. Discussion
First, we discuss the cause of the spectral change of the comb. In the previous section, we evaluated the spectral stability of two PM combs and two non-PM combs under free-running and phase-stabilized conditions. As a result, the spectral stability often appeared to be lower when feedback control was applied while no significant difference in the stability could be observed between the PM and non-PM combs. Figure 6 shows the measured cosine similarity variation of phase-stabilized PM Comb #1 and PM Comb #2.
Fig. 6. Measured cosine-similarity time series of the spectrum of a phase-stabilized comb, and the cosine-similarity time series of the spectrum of the comb calculated from the control amount time series to the pumping laser in each mode-locked laser and the degree of spectral change β1 (Table 2).
Then, the control amount of the injection current to the pump laser for the oscillator, which was simultaneously measured with the above cosine similarity, was converted to cosine similarity using Eq. (2) and the value of β1 in Table 2. In most cases, the results agreed well with the actual cosine similarities. This strongly suggests that this spectral change is due to the control of the pump-laser power for the oscillator. However, not all the spectral changes could be explained by the change in the amount of control for the pump power (e.g., Fig. 6(a)) and further investigation is required.
Next, we discuss the comb conditions needed to achieve high spectral stability in dual-comb spectroscopy. In dual comb spectroscopy, frep and fCEO stabilizations are required in most cases. The frep of the comb is often stabilized by controlling the resonator using PZT or EOM, but in many cases they are absorbed by the temperature control, and it is the oscillator temperature that fluctuates in the long term. In addition, since the PM comb did not clearly improve the spectral stability, we consider it is more important for dual-comb spectroscopy that the comb spectrum is insensitive to changes in oscillator temperature and pumping laser power for the oscillator than whether or not the system consists of PM fiber. However, this experiment was carried out in our laboratory, and the situation may be different in the field or in an extreme environment.
Although the results described in this study do not clarify how to fabricate combs whose spectra are insensitive to frequency control, it is possible to select suitable combs for dual-comb spectroscopy by evaluating the degree of spectral change in response to the amount of control using cosine similarity. Introducing evaluations such as the one performed in this study when fabricating combs would be helpful for high-precision dual-comb spectroscopy.
6. Conclusion
This paper introduced and experimentally applied cosine similarity as a quantitative measure of the spectral stability of a broadband optical frequency comb. The validity of the evaluation method using cosine similarity was demonstrated by measuring the spectral changes caused by the control imposed on the comb, such as the pumping power for a mode-locked fiber laser. The cosine similarity was then used to evaluate the spectral stability of PM and non-PM combs under free-running and phase-stabilized conditions. In conclusion, we suggested that reducing the spectral change with respect to the control amount is more important than maintaining polarization in terms of improving the spectral stability of the comb. We assume that it would be helpful to fabricate combs that are insensitive to the amount of control by introducing the evaluation shown in this paper. The evaluation of spectral similarity and temporal stability by cosine similarity is applicable to various types of broadband light including optical frequency combs and is also a general-purpose method that can be used in a wide range of fields.
Funding
Program on open innovation platform for industry academia co-creation (JPMJPF2015); Exploratory Research for Advanced Technology (JPMJER1304); Japan Society for the Promotion of Science (16K05009, 19H02610, 26800217).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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