1. Introduction

Continuous-wave (CW) lasers with a high frequency precision have played an important role in the progress of science and industry. They have become indispensable basic tools for applications such as precise atomic/molecular spectroscopy and interferometry-based length/shape measurement. Recently, these applications have demanded lasers with both high frequency precision and mode-hop-free continuous tunability over a wide frequency range. On the other hand, the recent development of optical frequency combs and frequency-stabilized lasers using atomic/molecular transitions has greatly improved laser frequency precision; a number of CW lasers with a wide mode-hop-free range have appeared. As a result, precisely and widely tunable laser systems (PW-TLSs) have been reported that simultaneously provide both a frequency precision exceeding 12 digits and continuous tunability over more than 10 GHz.

A PW-TLS is important for obtaining precise interferometric length measurements. In particular, in the International Avogadro Coordination project, whose aim is the re-definition of the kilogram in the International System of Units (SI), PW-TLSs have been employed to measure the diameter and volume of a $^{28}$Si-enriched monocrystalline silicon sphere [1–3] based on the phase shift method [4,5]. After the kilogram was re-defined in May 2019, and now the volume of the silicon sphere is being measured to realize the new definition of the SI kilogram [6]. The precision of the diameter measurement of the silicon sphere is considered to be limited by the diffraction effect of the laser beam near the sphere surface [2]. The evaluation and suppression of this diffraction effect requires a PW-TLS with both a wavelength shorter than the value of 633 nm used so far and a wider tunable frequency range.

A PW-TLS basically consists of a master and a slave laser. The master ensures frequency precision; the slave is phase-locked to the master so that the slave frequency can be widely tuned while maintaining the frequency precision of the master. For the precise volume measurement of the monocrystalline silicon sphere described above, in 2003, Kuramoto and Fujii used a laser phase-locked to an iodine-stabilized He-Ne laser as a master and swept the slave laser frequency over 20 GHz [1]. To the best of our knowledge, this was the first demonstration and application of PW-TLSs. They subsequently adopted a laser phase-locked to an optical frequency comb as a master to improve frequency precision and system robustness [3].

Other frequency sweep methods utilizing the broadband nature of the frequency comb have been proposed. Jost et al. phase-locked a 780 nm slave laser to a frequency comb and swept the slave frequency over several tens of GHz excluding some dead zones [7]. Schibli et al. developed a more sophisticated technique and continuously swept a 1064 nm slave frequency over more than 20 GHz while phase-locking it to the comb [8].

It is also effective to take advantage of the inherent mode-hop-free nature of the frequency comb. Washburn et al. pointed out that the slave frequency can be widely swept by using a frequency comb as the master and changing its repetition rate [9]. By using this technique, several groups have continuously swept a phase-locked 1542 nm slave frequency over more than 2 GHz to observe the Doppler profiles of acetylene absorption lines [10–12]; Park et al. have continuously swept an injection-locked 852 nm slave frequency over more than 1 GHz to observe a saturated absorption spectrum of cesium $D_2$ line [13]; but a frequency sweep exceeding 10 GHz has yet to be reported.

A frequency sweep approach has also been proposed that utilizes the wide modulation bandwidth of electro-optic modulators (EOMs). It is easy to sweep a modulation sideband of more than 10 GHz by changing the frequency applied to the EOM. Cole and Mohan demonstrated a power-lossless laser frequency sweep using injection locking and an EOM [14]. Inaba et al. swept an 894 nm slave frequency over 12 GHz using an EOM [15].

However, the above-mentioned methods present at least one of the following difficulties; (1) the need to detect and phase-lock the beat note at a high frequency such as 10 GHz, (2) the need to change frequency to detect and/or phase-lock the beat, (3) the need to change the repetition rate of the comb and/or to use an injection-locking technique, and (4) the fact that high optical power cannot be obtained. (1) and (2) degrade the robustness and reliability of the phase-locking; (3) makes the system and operation complicated; (4) significantly limits the scope of the applications.

In this study, we propose and demonstrate a novel way to overcome these difficulties. Our method achieves reliable beat detection at a fixed frequency, a robust and wide frequency sweep using an EOM, and small optical power loss, enabling the robust and precise continuous frequency sweep of the slave over more than 20 GHz at 852 nm. Furthermore, to improve the precision of the volume measurement for a monocrystalline silicon sphere, we demonstrate a continuous frequency sweep of more than 40 GHz at 426 nm by generating the second harmonic with a WG-PPLN.

2. Experimental setup

Figure 1(a) shows the 426 nm CW laser system developed in this study. Here we employ an 852 nm interference-filter-based external-cavity diode laser (IF-ECDL) as the master laser and an 852 nm distributed Bragg reflector (DBR) laser as the slave laser. The IF-ECDL (master) is robust to environmental fluctuation and has a narrow linewidth, which facilitates the stabilization of the laser frequency to frequency references including frequency combs, while the mode-hop-free tunable range is somewhat narrow (typically less than 10 GHz). On the other hand, the DBR laser (slave) has a wide mode-hop-free tunable range exceeding 100 GHz and a high output power of more than 100 mW, which enables it to be used for many applications and for second harmonic generation directly. The slave frequency is stabilized by controlling it so that the beat frequency between the master and an EOM-generated sideband of the slave is constant as shown in Fig. 1(b). The carrier frequency of the slave can be swept by sweeping its modulation frequency, with the sideband of the slave locked to the master.

 

Fig. 1. (a) Schematic of the experimental setup. IF-ECDL: interference-filter-based external-cavity diode laser, DBR laser: distributed Bragg reflector laser, OI: optical isolator, SMF: single-mode fiber, PMF: polarization-maintaining fiber, WG-PPLN: waveguide-type PPLN, EOM: electro-optic modulator, PD: photodetector, TEC: thermo-electric cooler. (b) Frequency relation between the master (IF-ECDL) and slave (DBR laser).

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Here we provide details of the IF-ECDL used as the master laser; the laser is home-made, and its cavity consists of a half mirror (output coupler) with a reflectivity of 20% and a semiconductor gain chip with a high-reflection coating on one facet. A bandpass optical interference filter with a full width at half maximum (FWHM) of 0.5 nm is inserted in the cavity. The laser wavelength can be roughly tuned by adjusting the filter angle with respect to the optical axis [16,17].

Next, we describe in detail the DBR laser that we used as the slave laser; the laser chip is commercially available (Photodigm, PH852DBR240T8) and is built in a TO-8 package with a thermo-electric cooler. According to the data sheet, the maximum power exceeds 240 mW, and the wavelength sweep range is 851.9-852.5 nm. We confirm that the laser frequency can be swept over more than 130 GHz without any mode hopping by sweeping the chip temperature.

The output beam of the master laser (IF-ECDL) is passed through an optical isolator to suppress the back reflection and coupled into a single-mode fiber; the beam is mixed with the slave (DBR laser) in the first fiber coupler to detect the beat signal. On the other hand, the output beam of the slave laser is also passed through an isolator and coupled into a polarization-maintain fiber; the coupled beam is divided into two parts at a ratio of 99% and 1% at the second fiber coupler. 1% of the slave is passed through the in-line EOM to generate sidebands in the frequency domain and is mixed with the master in the first fiber coupler to detect the beat signal. The modulation frequency of the slave after passing through the EOM is $f_{\mathrm {mod}}$; sidebands are generated on both sides of the slave carrier frequency in the frequency domain. To detect the beat signal, which corresponds to $f_{\mathrm {beat}}$ in Fig. 1(b), between the master and a sideband of the slave, a photodetector is irradiated with the beam mixed in the first fiber coupler. On the other hand, 99% of the slave beam is coupled into a WG-PPLN with a poling period of 9.6625 $\mu$m; a 426 nm second harmonic beam is generated under a 3rd-order phase matching condition [18].

Figure 1(b) shows the frequencies of the master (IF-ECDL), and the slave (DBR laser) with a carrier and the $\pm$1st-order sidebands. The carrier frequency of the slave $\nu _{\mathrm {slave}}$ can be expressed as

In this scheme, we phase-lock $f_{\mathrm {beat}}$ so that it is constant even if $f_{\mathrm {mod}}$ changes when the laser frequency is swept. The beat signal with the frequency $f_{\mathrm {beat}}$ is detected with a photodetector; the detected beat signal is filtered, amplified, and frequency-divided by 200; its phase is compared with the phase of the reference frequency of 1 MHz provided from the frequency synthesizer, using a digital phase frequency comparator. To phase-lock $f_{\mathrm {beat}}$, the error signal obtained by the signal processing is fed back to the injection current and thermo-electric cooler of the slave laser through the loop filter. Here the $-$1st-order sideband of the slave is phase-locked to the master at an offset frequency of 200 MHz.

In this situation, we can sweep $\nu _{\mathrm {slave}}$ by sweeping the modulation frequency $f_{\mathrm {mod}}$. In this study, the slave was continuously swept over 20 GHz at 852 nm (40 GHz at 426 nm) by sweeping $f_{\mathrm {mod}}$ from 0.2 to 20 GHz. The frequency sweep step was set at 5 MHz for the 852 nm fundamental light (10 MHz for the 426 nm second harmonic light); the time interval between each step was set at 50 ms, which corresponds to a frequency sweep rate of 100 MHz/s for the 852 nm fundamental light (200 MHz/s for the 426 nm second harmonic light). We could generate a sideband with sufficient power up to $f_{\mathrm {mod}} = 20$ GHz by adjusting the modulation power depending on $f_{\mathrm {mod}}$ despite using an EOM with a cutoff frequency of 10 GHz. The sweep range of $f_{\mathrm {mod}}$ was limited by the maximum output frequency of the synthesizer used in this study.

3. Results

Figure 2(a) shows the spectrum of the beat signal between the master and slave lasers detected with a photodetector. The beat signal between the master and the carrier of the slave and between the master and the $\pm$1st sidebands of the slave are observed. $f_{\mathrm {beat}}$, the beat frequency between the master and the $-$1st sideband of the slave, are phase-locked at 200 MHz. $f_{\mathrm {mod}}$, the modulation frequency of the slave, is swept from 0.2 to 20 GHz. Figure 2(a) shows the beat spectrum when $f_{\mathrm {mod}}$ was set at 0.2 GHz.

 

Fig. 2. (a) RF spectrum of the beat signals between the master laser (IF-ECDL) and the carrier and sidebands of the slave laser (DBR laser) before filtering and amplification. $f_{\mathrm {beat}}$ was phase-locked to 200 MHz and $f_{\mathrm {mod}}$ was set at 200 MHz. *Spurious signals originating from an unknown modulation mixed into the control electronics for the master laser (IF-ECDL), **a spurious signal originating from an unknown radio wave mixed into the spectrum analyzer. (b) In-loop $f_{\mathrm {beat}}$ spectrum after filtering and amplification. (c) Allan deviation of the in-loop $f_{\mathrm {beat}}$.

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Figure 2(b) shows the in-loop beat spectrum. The frequency of the in-loop beat is counted with $\pi$-type dead-time-free counters [19] at a gate time of 1 s to evaluate the performance of the phase lock. Figure 2(c) shows the Allan deviation of the in-loop $f_{\mathrm {beat}}$. The Allan deviation decreases in proportion to $1/\tau$ ($\tau$ is the averaging time), which indicates that the phase lock functions properly. The relative Allan deviation of the slave frequency is $2.8 \times 10^{-14}/\tau$, which is calculated by dividing the Allan deviation of $f_{\mathrm {beat}}$ by a laser frequency of 352 THz corresponding to a wavelength of 852 nm. Note that in most cases the absolute frequency stability of the slave is limited by that of the master. In addition, the true relative frequency stability obtained with out-of-loop measurement is worse than that obtained with in-loop measurement because the slave laser suffers some additional frequency fluctuations caused by noise sources such as optical fiber and the frequency synthesizer that provides the modulation signal.

Next, we evaluated a 426 nm laser source using a WG-PPLN. Figure 3(a) shows the relation between the input 852 nm laser power and the output 426 nm second harmonic power. The conversion efficiency $\eta$ was 14 %/W when the relation between the second harmonic power $P_{2\omega }$ and fundamental power $P_\omega$ was expressed as $P_{2\omega } = \eta {P_\omega }^2$. Figure 3(b) shows the WG-PPLN temperature dependence of the second harmonic power. The allowable temperature range (full width at half maximum) for phase matching was 0.3 $^\circ$C, which is very narrow due to the use of 3rd-order phase matching. The red curve in Fig. 3(c) shows the 852 nm input laser frequency dependence of the 426 nm second harmonic power. The allowable frequency range of the 852 nm input laser was 8 GHz. Therefore, the WG-PPLN temperature must be precisely adjusted to obtain a sufficiently stable and high second harmonic power when the laser frequency is swept. In advance, we measured the temperature at which the second harmonic power was maximized by sweeping the frequency of the input laser to the WG-PPLN over 20 GHz (stepping frequency: 100 MHz). The WG-PPLN temperature was feedforward-controlled so that the second harmonic power was always maximized when the 852 nm laser frequency was swept. The blue curve in Fig. 3(c) shows the 426 nm laser power when the laser frequency was swept while employing feedforward temperature control. The 426 nm output power fluctuated up to 15% even when employing the WG-PPLN temperature control. More precise stabilization of the output power requires an additional power controller such as an acousto-optic modulator or a variable optical attenuator.

 

Fig. 3. Second harmonic (SH) power from the WG-PPLN at 426 nm. (a) Input power dependence; the red diamonds show the relationship between the fundamental power input into the coupling optical fiber and the SH power output from the WG-PPLN; the black curve is a fitting curve when a quadratic function is adopted as the fitting function. (b) WG-PPLN temperature dependence when the fundamental laser frequency is constant. (c) Input 852 nm laser frequency dependence; the red and blue lines show the output 426 nm laser power without and with temperature correction, respectively.

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We demonstrated a continuous frequency sweep with the 426 nm laser. To confirm that neither mode hopping nor lock interruption occur during the frequency sweep, the 426 nm laser output from the WG-PPLN was led to a Michelson interferometer as shown in Fig. 4(a); the output power from the interferometer was measured as an interference signal while sweeping the laser frequency. Figure 4(b) shows the interference signal; the obtained signal is continuous and periodic with a repetition period of 4.5 GHz, which shows that the 426 nm laser frequency is continuously swept over 40 GHz without any mode hopping or lock interruption. The period of 4.5 GHz is consistent with the optical path difference between the two arms in the interferometer (70 mm $\pm$ 10 mm), which was measured with a ruler. We assume the distortion of the interference signal to be due to the optical path length fluctuation in the Michelson interferometer caused by air fluctuation.

 

Fig. 4. (a) Setup and (b) obtained signal of Michelson interferometer.

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4. Discussion and conclusion

Our proposed wide-frequency-sweep technique has several advantages. First, the beat frequency can be fixed for detection and/or phase-locking. Therefore, beat detection at a high frequency such as 10 GHz is not required; a beat signal with a high signal-to-noise ratio can be easily obtained; a narrow RF bandpass filter can be used, which contributes to robust and reliable phase-locking. Second, no optical frequency comb is required for the frequency sweep; even if we use a comb we can fix its repetition rate. This is a particularly important issue when the comb is simultaneously used for other applications. Third, we are able to use the carrier component of the slave laser for applications although we use an EOM; therefore, this technique does not require a large power loss as is the case when using a sideband component for applications. This is also an important feature for many applications including spectroscopy and interferometry. In addition, we can use cost effective and robust RF devices such as amplifiers and filters at less than several hundred MHz with many specification options.

There are several methods for providing a slave laser with an absolute frequency; for example, phase-locking the master laser to a mode of a frequency comb or using a frequency-stabilized laser with an atomic/molecular absorption line; it may also be possible to directly lock the master laser to an atomic/molecular absorption line. The mode number must be determined when using the optical frequency comb. There are a number of approaches for determining the comb mode number. Methods including one using a precise wavemeter, and one using an additional comb [20,21] are available. In this study, we described a method using a frequency comb and a cesium $D_2$ line to determine the mode number of the comb. We tuned the master frequency to the $F = 3 \to 2$ transition ($F$ is total angular momentum) of the cesium $D_2$ line by observing the saturated absorption spectrum shown in Fig. 5 and phase-locked the master frequency to the nearest comb mode. The comb mode number can be specified from the $F = 3 \to 2$ transition frequency $351\:730\:549.61(11)$ MHz [22] since the frequency uncertainty is much less than the repetition rate of the used comb ($\sim$88 MHz).

 

Fig. 5. (a) Setup and (b) obtained signal of saturated absorption spectroscopy of cesium $D_2$ line. Hyperfine components are labeled as a: $F = 3 \to 2$, b: $F = 3 \to 3$, and c: $F = 3 \to 4$ transition, and their cross-over components are labeled as a-b, a-c, and b-c.

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In conclusion, we developed a CW laser system capable of a continuous sweep over a wide frequency range of 40 GHz with high frequency precision at 426 nm. We proposed and demonstrated the new frequency sweep scheme using an EOM. A stable optical power was achieved over the entire sweep range by feed-forward controlling the WG-PPLN temperature to satisfy the phase matching condition according to the laser frequency. Furthermore, we confirmed that we realized a 40 GHz continuous sweep without any mode hopping or lock interruption by observing the interference signal of a Michelson interferometer. This technique has practicality and precision for the wide and continuous frequency sweep of a laser; it has the potential to play an important role as a universal scheme in many applications.