2.1.

4-GHz Optical Frequency Comb

Figure 1 shows a schematic diagram of the 4-GHz OFC configuration, the highlight of which is the $\mathrm{Yb}:{\mathrm{Y}}_2{\mathrm{O}}_3$ ceramic Kerr-lens mode-locked laser. The laser configuration follows that of our previously developed 6-GHz $\mathrm{Yb}:{\mathrm{Lu}}_2{\mathrm{O}}_3$ ceramic laser.²³ The resonator is a bow-tie ring cavity with two concave mirrors ($r=15\mathrm{mm}$) and two plane mirrors, and the net cavity length was set to 46 mm. Both concave mirrors were dichroic mirrors (DMs) coated for high reflection (HR) at 1080 nm and high transmission at 976 nm. The plane mirrors were both HR-coated for 1080 nm. One of them was mounted on a piezoelectric actuator to allow phase stabilization of the repetition rate, whereas the other was chosen to be a chirped mirror (CM) to compensate the intracavity dispersion. The total group delay dispersion of the oscillator was estimated to be $-100{\mathrm{fs}}^2$, which allowed sufficiently high peak powers to induce the Kerr-lens effect. The gain medium, a 2-mm-thick noncoated 3 at. % $\mathrm{Yb}:{\mathrm{Y}}_2{\mathrm{O}}_3$ ceramic, was mounted on a copper plate and placed between the two concave mirrors at the Brewster angle. The ceramic gain medium was pumped by a single-mode-fiber coupled fiber-Bragg-grating stabilized 976-nm laser diode with a maximum power of 1 W. The folding angles at the concave mirrors were adjusted to 11 deg to compensate the astigmatism caused by the Brewster-placed ceramic. The pump beam was collimated then focused down to the center of the ceramic by a pair of singlet lenses ($f=25$ and 40 mm, respectively). The CM had a transmittance of 0.04% and was used as the output coupler. After optimization of the cavity alignment for continuous-wave (cw) operation, by adjusting the position of one of the concave mirrors, stable Kerr-lens mode-locked operation was obtained. In the ring cavity, both clockwise and counterclockwise oscillations were possible for cw operation, and two output beams were observed from the output coupler. When the laser was mode-locked, one of the output beams disappeared and the output power in the other would increase, indicating that a single oscillation direction was permitted inside the cavity for mode-locked operation. Under the mode-locked condition, the pulse spectrum was centered at a wavelength of 1078 nm with a FWHM of 7 nm and the Fourier transform-limited pulse duration was 174 fs. The optical spectrum is shown in Fig. 2. When the pump power was 700 mW, an output power of 20 mW was obtained and stable mode-locking was maintained for several days. The stabilization of each single mode of a frequency comb was achieved via phase-locking both the carrier-envelope offset ($f_0$) and repetition frequencies ($f_{\mathrm{rep}}$). Typically, in the case of the carrier-envelope offset frequency, phase-locking is achieved via the self-referencing technique, which consists in part of generating an octave-spanning spectrum using a highly nonlinear fiber. However, repetition rates as high as 4 GHz prevent such spectral broadening, therefore, stabilization was carried out by phase-locking the comb modes to those of a 250-MHz Yb-fiber OFC whose repetition rate was one sixteenth that of 4 GHz. Both repetition frequencies were phase-locked to rf standards, thus keeping the frequency ratio of 16:1, and in this way, the observed beat signal between the 250-MHz OFC and the 4-GHz oscillator would correspond to the difference between the carrier-envelope offset frequencies as shown in Fig. 3. To achieve this, the beams from the two OFCs were coupled into a 50:50-fiber combiner, and the beat signal was detected at the $f_{\mathrm{beat}}$ detector. When the repetition rate of the 250-MHz OFC was one-Nth ($N=16$ in our experiment) of the 4-GHz OFC repetition rate, the beat signals between all modes of the two OFCs were the same over the entire overlapping region of the combs’ spectra.

Fig. 1

The experimental apparatus of the 4-GHz OFC. PMF, polarization-maintaining fiber; DM, dichroic mirror; HR, high-reflection mirror; CM, chirped-coated mirror; HWP (QWP), half (quarter)-wave plate; gain medium, 2-mm-thick; noncoated 3 at. % $\mathrm{Yb}:{\mathrm{Y}}_2{\mathrm{O}}_3$ ceramic; PBS, polarization beam splitter; VCO, voltage-controlled oscillator; Amp., RF amplifier; $f_{\mathrm{rep}}$ ($f_{\mathrm{beat}}$) detector: 25-GHz (2-GHz) InGaAs photodetector; and YDFA, Yb-doped fiber amplifier.

Fig. 2

The optical spectrum of the 4-GHz OFC. The spectrum width was 7 nm at the center wavelength of 1078 nm.

Fig. 3

Beat signals between two OFCs with different repetition rates of $f_{\mathrm{rep}1}$ and $f_{\mathrm{rep}2}$. (a) A simplified schematic diagram of the experiment, where OFC1 is the 4-GHz comb and OFC2 is the 250-MHz one. (b) Optical frequency domain representation of a spectrum, as would be measured by an optical spectrum analyzer, of the interference between the modes of the 4-GHz OFC and 250-MHz OFC. (c) Is the same as (b), only it is the rf domain representation of a spectrum as would be measured by an electronic spectrum analyzer.

The 4-GHz repetition frequency was locked to a Rb-clock referenced rf standard using a 25-GHz-bandwidth InGaAs detector ($f_{\mathrm{rep}}$ detector) and an analogue locking circuit. An acousto-optic modulator (AOM) and a phase-locked-loop circuit were used to phase-lock the beat signal to the reference frequency (chosen here as 173 MHz). The AOM was mounted in a double-pass configuration to avoid shifts in the beam pointing when the frequency was tuned. An Yb-doped fiber amplifier (YDFA) was placed after the AOM to increase the SNR of the beat signal for efficient detection. The output power as measured at “4-GHz OFC output” in Fig. 1 was $\sim 50\mathrm{mW}$.

Figure 4 shows the beat signal when the 4-GHz frequency comb was free-running (a) and when it was phase-locked (b). Although the locking bandwidth was limited to $\sim 100\mathrm{kHz}$ by the modulation bandwidth of the voltage-controlled oscillator (VCO) used with the AOM, a higher bandwidth (over a MHz) could be obtained by simply replacing the VCO with one that has a higher modulation bandwidth. Higher phase-locking bandwidths and higher amplitudes of the frequency modulation are required for frequency combs with GHz repetition rates over those with sub-GHz repetition rates, since cavity length fluctuations result in larger frequency shifts of the comb modes. In order to maintain the phase-locking of the repetition rate for a longer time still, the oscillator could be temperature controlled, however, this was not done here. The modulation bandwidth and the fact the cavity was not temperature controlled restricted the long-term and phase stabilities of the system, but under the standard laboratory conditions, the OFC was stable over multiple days without the need for any stabilization.

Fig. 4

Spectra of beat signals measured with $f_{\mathrm{beat}}$ detector. (a) Free-running signal measured with a resolving bandwidth (RBW) of 100 kHz. (b) Phase-locked beat signal acquired with an RBW of 1 kHz.

2.2.

Multipass Spectrograph with a Frequency Resolution of Sub-GHz

A multipass spectrograph, as depicted in Fig. 5(a), was assembled using a transmission grating ($1740\mathrm{g}/\mathrm{mm}$) with a high efficiency (90% at $\lambda =1080\mathrm{nm}$) and a large size ($180\mathrm{mm}\times 40\mathrm{mm}$). The transmission grating (made by Canon Inc.) was fabricated by optical lithography means and further details can be found in Ref. 24. Owed to the multipass configuration, the spectrometer had a relatively small footprint ($\sim 0.5\mathrm{m}\times 1\mathrm{m}$). The stabilized output of the 4-GHz frequency comb was guided to the spectrograph configuration via a polarization-maintaining single-mode fiber (PMF), the core of which ($6\mu \mathrm{m}$ mode-field diameter) served as an incident slit for the spectrometer. The light was collimated to a line shape ($\sim 1\mathrm{mm}\times 25\mathrm{mm}$) by two perpendicularly placed cylindrical lenses [with focal lengths of 15 mm (CYL 1, LJ1636L2-B, Thorlabs) and 150 mm (CYL 2, ACY254-150-B, Thorlabs), respectively]. The light incident on the grating was diffracted multiple times using mirrors (HR 1, 2, and 3) until the desired resolution was obtained. Figures 5(b) and 5(c) illustrate the beam path of a simple three-pass configuration. In our experiments, the number of diffractions was extended to 7 with this same configuration. The numbers (1 to 6) point to the order in which the beam is being diffracted or reflected after collimation at CYL 2. The line-shaped light from CYL 2 was first diffracted by the grating (1), then reflected by HR 2 (2). The light was diffracted again at the grating (3). The mirror HR 1 was slightly tilted in the vertical direction, so that after the second diffraction at point (3), the beam was $\sim 1\mathrm{mm}$ below the first one (1). The diffracted light was reflected at HR 2 (4); then diffracted at (5). In Figs. 5(b) and 5(c), the three-times diffracted light was reflected off of HR 3 (6) and directed toward CYL 3. The multiply diffracted light was then focused onto a CCD camera ($3312\times 2488$ pixels, 14-bit resolution, IGV B3320M, IMPERX) by two perpendicularly placed cylindrical lenses [with focal lengths of 500 mm (CYL 3, specially designed by OptoSigma) and 75 mm (CYL 4, ACY254-075-B, Thorlabs), respectively]. To reduce aberrations, three of the cylindrical lenses used were doublets (achromatic lenses). Upon the CCD surface, the diameter of the focused beam was $25\mu \mathrm{m}$ and the beam spot separation was $150\mu \mathrm{m}$ for a seven-pass configuration. With this configuration, a 610-MHz frequency resolution was calculated ($R=f/\mathrm{\Delta }f=\mathrm{450,000}$), and each longitudinal mode of the 4-GHz OFC was clearly resolved at the CCD camera, as shown in Figs. 5(d) and 5(e). The presence of weak beam spots in between each main one was most likely due to aberrations of the achromatic cylindrical lenses. Typically, the assembly of an achromatic cylindrical lens is harder than that of a standard achromatic one, as they are more sensitive to the axis of alignment. Since the size of the collimating and focusing lenses used was limited, diffraction effects occurred at these lenses. These aberrations and diffraction effects contributed to a focal spot size that is slightly larger than the anticipated diffraction-limited one of $\sim 20\mu \mathrm{m}$. Lenses with larger diameters could help reduce these effects and improve the frequency resolution.

Fig. 5

(a)–(c) The experimental configuration of the multipass spectrometer. (b), (c) The top and side views, respectively, of the beam path. The beam was diffracted or reflected from 1 to 6 at the grating or the high-reflection mirrors (HR 1, 2, and 3). In this figure, the beam was diffracted 3 times at the grating. (d), (e) The resolved comb modes of the 4-GHz OFC by the multipass spectrograph when the center wavelength was set to 1077.5 nm. (e) The expanded image.

The spectrometer had a bandwidth of 1 nm centered at the wavelength of 1077.5 nm and a throughput of over 10% was achieved with the resolution of 610 MHz. The center wavelength could be changed by rotating the grating and HR 1. The frequency resolutions and resolving powers are detailed in Table 1. Although the setup described in Fig. 5 enabled the use of light having undergone an odd number of diffractions, an even number of diffractions was achieved by simply changing the spectrograph configuration.

Table 1

Number of diffractions and corresponding measured resolution.