1. Introduction

Stabilization of the frequency comb of a mode-locked femtosecond laser enables one to calibrate optical frequencies with direct traceability to the radio-frequency time standard [1,2]. This advance prompted the attempt of generating optical frequencies with high precision by exploiting the frequency comb as the reference ruler, as first demonstrated by phase-locking a tunable source to the stabilized comb of a Ti:Sapphire femtosecond laser [3–5]. A more accurate method of optical frequency generation was afterwards introduced by extracting a single mode of the desired frequency from the frequency comb with subsequent power amplification by means of optical injection locking to a laser diode [6,7]. This straightforward use of the frequency comb permits the generated frequency signal to inherit the superb linewidth and absolute frequency position of the original comb. However, the problem is that the optical injection locking for power amplification suffers low stability. This is because the locking range of a laser diode is narrowly confined, with its center frequency drifting with environmental temperature change [8–10], Moreover, the extraction of a single mode using a diffractive grating is vulnerable to vibration since the frequency comb is densely populated with closely spaced modes. Here in this investigation we demonstrate an all-fiber-based apparatus built to generate near-infrared frequencies directly from an Er-doped fiber femtosecond laser with high immunity to vibration. Furthermore, the fragile process of optical injection locking is stabilized by incorporating a special scheme of phase lock-in control such that the locking range of the amplifying laser diode is continuously adjusted to fit to the extracted frequency mode all the time even in the presence of severe temperature change.

2. All-fiber-based optical frequency generation

Figure 1(a) shows the hardware configuration of the all-fiber-based apparatus constructed in this study. An Er-doped fiber femtosecond laser (C-Fiber, Menlo Systems GmbH) was used to produce a train of short pulses of ~100 fs duration at a 100 MHz repetition rate with an average power of ~20 mW. The frequency comb of the fiber femtosecond laser having a 50 nm bandwidth centered at 1550 nm in wavelength was stabilized by resorting to well-established techniques [11]. Specifically, the repetition rate was phase-locked to a standard Rb clock by adjusting the fiber cavity length using a piezoelectric actuator. The carrier-offset frequency was monitored through an f-2f self-referencing interferometer using a highly nonlinear fiber for spectral broadening [12] and also a periodically poled lithium niobate (PPLN) crystal for frequency doubling. The pumping power to the fiber cavity was controlled for the carrier-offset frequency to be phase-locked to a predetermined value.

Extraction of a wanted mode out of the stabilized frequency comb was made by devising a composite filtering scheme that combines a scanning Fabry-Perot fiber etalon with a fiber Bragg grating in sequence. The used Fabry-Perot etalon (FFP-I, Micron Optics) has a ~50 GHz free spectral range with a high finesse of 200, providing multiple transmission windows equally distanced over the entire spectral range of the frequency comb as illustrated in Fig. 1(b). Each window offers a narrow filtering bandwidth to transmit only three consecutive modes; a central mode with higher amplitude and two side modes with attenuated amplitudes. Next, the fiber Bragg grating (DWDM101C59, AC Photonics) provides only one transmission window of a relatively broad bandwidth of ~100 GHz. Thus, adjusting the spectral position of the Bragg grating’s window permits the selection of only a particular transmission window of the Fabry-Perot etalon that carries the wanted frequency in the central mode as illustrated in Fig. 1(b).

 

Fig. 1. Apparatus designed for optical frequency generation. (a), System block diagram (optical signals in red and electric signals in blue). (b), Mode extraction through two sequential filtering steps prior to final optical injection locking. Each of the encircled numbers, ①–④, indicates the location of the corresponding frequency spectrum. FBG: fiber Bragg grating, f: frequency, f₀: carrier-offset frequency, f_r: repetition rate, PC: polarization controller, PD: photodetector, SFPF: scanning Fabry-Perot filter.

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The Fabry-Perot filter is initially adjusted to maximize the transmitted intensity of the selected mode. Then it is fixed with temperature control within a tight range of 0.01 °C corresponding to the frequency variation of 17 MHz. This guarantees the selected mode to be consistently positioned around the center frequency of the Fabry-Perot filter. Finally transmitted three modes yield weak individual power, with the central mode having ~40 nW. Thus, power amplification follows by means of optical injection locking to a laser diode [6,7,13]. Specifically, the three modes contained in the finally selected window of the Bragg grating are fed simultaneously to a laser diode (FOL15DCWB-A81-W1530, FITEL) of distributed-feedback type with its built-in isolator removed beforehand for easy entrance of external light into the diode cavity. Among the three modes, only the central mode is selectively locked and amplified with a gain of 57 dB to a power of 20 mW. Due to their relative low seeding power, the side modes are defeated by the central mode without significant power enhancement [8,14]. This discriminatory nature of optical injection locking consequently leads to efficient suppression of the side modes, thereby enabling the achievement of the final task to single out only the wanted frequency signal with adequate power amplification.

3. Stabilization control of optical injection locking

The aforementioned feat of optical injection locking is guaranteed only when the laser diode is precisely tuned to embrace the injected central mode within its locking range. As depicted in Fig. 2(a), the locking range f_LR is narrowly confined to a small spectral slot of ~1 GHz around the free-running frequency of the laser diode [9,10]. This free-running frequency determined by the diode cavity length is sensitive to environmental temperature change, as is the spectral position of the locking range f_LR [15]. The electric current input to the diode enables the control of the free-running frequency by varying the cavity length to minimize the frequency offset Δf between the free-running emission and the central mode. However, the instantaneous amount of Δf to be fed back to control the input current is not simply identified since the free-running emission disappears upon the occurrence of injection locking, while the diode gives all its output power to the injected central mode.

As illustrated in Fig. 2(b), our stabilization scheme exploits the phase delay that would be induced in the central mode by its imperfect resonance due to the frequency mismatch within the given diode cavity length. Theoretical analysis [15] indicates that the phase delay varies with Δf following an arcsine pattern as plotted in Fig. 2(c). The two side modes injected into the laser diode also experience the same phenomenon of phase delay but in different amounts depending on their positions within the locking range. The phase delay is quantified by observing the beat signals produced between the central mode and side modes at the heterodyne frequency of f_r at the exit of the laser diode using an electric circuit comprised of a photodetector, a f_r-reference signal mixer and a low-pass filter as shown in Fig. 2(b). The resulting signal represents the intensity sum of two beat signals, and its dependence on Δf is shown in Fig. 2(d). This intensity signal is an even function of Δf, reaching its maximum when Δf=0. Finally, the feedback signal is obtained by modulating the diode input current at a particular frequency of f_PM with the subsequent lock-in detection of the beat intensity signal [16]. The feedback signal finally obtained is shown in Fig. 2(e), which is in fact the derivative of the intensity variation of Fig. 2d with respect to Δf. This feedback signal is an odd function of Δf and provides information on the magnitude and also sign of Δf, enabling adequate adjustment of the input current to the laser diode in order to reduce Δf to zero. The well-established PID (proportional-integral-derivative) control method is adopted to determine the input current Ifeedback such that the diode cavity length is always appropriate for the central mode for proper injection locking.

 

Fig. 2. Stabilization control of optical injection locking. (a) Tuning of the diode locking range to three injected modes. The central mode is the main frequency signal to be amplified and other two side modes are also injected to obtain the phase control signal. (b) Block diagram for phase stabilization control. (c) Phase delay of the injected signal after optical injection locking. (d) Intensity variation of beat signals. (e) Phase control signal induced by current modulation. Δf: locking offset, f_LR : diode locking range, f_PM : phase modulation frequency, f_r : repetition rate, I_PM : modulation current, I_feedback : feedback current, LPF: low-pass filter, PD: photodetector, PID: proportional-integral-derivative.

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We present an overview of the main steps taken to conduct the phase-stabilization control of the optical injection locking process. The phase delay shown in Fig. 2(c) is analytically given as [15]

Then the electric wave of the central mode combined with the two side modes at the outlet of the laser diode can be expressed in a normalized form of

where, i≡√-1 and α represents the amplitude ratio of the two side modes to the central mode. The combined wave of Eq. (2) leads the intensity to be expressed as the sum of the four separate components:

where, R(Δf,f_r)=exp[−iφ(Δf)+iφ(Δf−f_r)]+exp[iφ(Δf)−iφ(Δf+f_r)]. Only the sin(2πf_rt)-related term is isolated by adopting the radio frequency mixing technique as depicted in Fig. 2(b). The corresponding partial intensity is proportional to the imaginary part of R(Δf, f_r), and it is an even function of Δf as illustrated in Fig. 2(d).

Now we assume that f_r is much smaller than f_LR. Then, using the approximation of exp(x) ≈1+x for |x|<1, the partial intensity can be rewritten as

Finally, the feedback signal shown in Fig. 2(e) that was obtained by the lock-in modulation of Fig. 2(b) is derived by taking the derivative of Eq. (4) with respect to Δf as such

This can be further simplified, for a small Δf, into the linear equation of

The above final result of the feedback signal clearly indicates a linear dependence on Δf for a given set of f_r and f_LR, which enables effective phase stabilization of the optical injection locking process.

4. Experimental verifications

For verification of our stabilization control, the frequency output signal finally generated was made interfered with the original frequency comb, while the resulting beats were observed using an rf spectrum analyzer. As shown in Fig. 3, when injection locking was properly activated, the output signal was dominated by the amplified central mode, thereby producing no beats in the vicinity of the mode spacing frequency of f_r. Otherwise, the free-running frequency emitted from the laser diode would symmetrically produce two beats around f_r and those beats floated with temperature change. Next, the linewidth profile of the generated output signal was measured. For that measurement, a small portion of the femtosecond laser source was diverted first and transmitted through a spectral band-pass filter of 0.8 nm bandwidth to take out a partial comb containing the original frequency mode corresponding to the generated frequency output signal, and the obtained partial comb was frequency-shifted by ~40 MHz using an acousto-optic modulator (AOM). Then the frequency output was made interfered with the partial comb, and the resulting beat signal was observed using a high resolution rf spectrum analyzer (E4440A, Agilent) that provides a 1 Hz resolution bandwidth. This measurement confirmed that the linewidth profile of the output signal has a narrow full-width-at-half-maximum (FWHM) of less than 1 Hz as shown in Fig. 4.

 

Fig. 3. Verification of optical injection locking by monitoring the output signal emitted from the laser diode using an rf spectrum analyzer. When optical injection locking occurs, the output signal is dominated by the amplified central mode to produce only a single beat f_r with the original frequency comb. Otherwise, the free-running emission from the laser diode makes another beat f_beat, which drifts with temperature change (inset). f_r: repetition rate of the frequency comb.

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Fig. 4. Linewidth measurement of the generated frequency output signal using a high resolution rf spectrum analyzer of 1 Hz resolution bandwidth (RBW). The output signal was interfered with a partial comb containing the original frequency mode of the output signal. The partial comb was frequency-shifted using an acousto-optic modulator by an amount of 41.00160 MHz to observe the linewidth profile in the rf domain. The full-width-at-half-maximum (FWHM) was measured to be less than 1 Hz.

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Finally, the temporal stability of the beat signal obtained in Fig. 4 was tested using an rf frequency counter continuously at a sampling rate of 0.5 s over a long period of 24 hours. Environmental temperature and vibration were not controlled with the intention of evaluating the overall robustness of our apparatus in normal laboratory environment. During the testing, room temperature underwent a variation of ±5 °C and the apparatus was placed on a standard optical table without anti-vibration air support. Figure 5(a) shows an actual time trace of frequency fluctuation, which yields a standard deviation of 3.4 Hz. Breakdown due to the failure of injection locking was not observed at all during the whole monitoring period, which is not the case when the phase-stabilization control is not applied. The Allan deviation was also worked out as plotted in Fig. 5(b), which is 2.3 parts in 10¹⁵ at 10 s averaging and reduces to one part in 10¹⁶ at 1000 s averaging. Note that the original frequency-stabilized comb is subject to a fractional stability of 9 parts in 10¹³, which is in fact two orders of magnitude worse than the measured temporal stability. This comparison confirms that the whole procedure of our optical frequency generation causes no substantial degradation in frequency stability, thereby permitting the absolute frequency position of the original comb to be maintained in the generated frequency output.

 

Fig. 5. Stability test result of the generated output signal. (a) Time trace of frequency fluctuation at a sampling rate of 0.5 s over a period of 24 hours. (b) Allan deviations of frequency instability with varying average time.

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5. Conclusion

In conclusion, the all-fiber-based apparatus built in this study allows us to generate near-infrared optical frequencies of an absolute uncertainty of 9 parts in 10¹³ from the stabilized comb of an Er-doped fiber femtosecond laser. The output frequency signal provides a narrow linewidth of less than 1 Hz and a net optical power of 20 mW after amplification through phase-stabilized injection locking to a laser diode. The long-term frequency stability over a period of 24 hours reaches 2 parts in 10¹⁵ at 10 s averaging, showing high immunity to environmental disturbance. Our method is precise and robust and, therefore well suited for on-site applications or space missions of near-infrared frequency calibration, spectroscopy, and optical clocks.