We propose a novel magneto-optical approach for the repetition frequency stabilization of optical frequency combs. We developed a Yb:fiber mode-locked laser with a fiber-based magneto-optic modulator used to stabilize one of the longitudinal modes to an optical reference with sub-hundred mrad residual phase noise. This modulator does not induce mechanical resonances and as such has the potential to achieve much broader feedback bandwidths than conventional modulators used for cavity length stabilization.
© 2017 Optical Society of America
Optical frequency combs (OFCs) have become invaluable tools for various applications including frequency metrology, calibration of astronomical spectrographs and gas sensing [1, 2]. For such applications, the two degrees of freedom of the OFC, namely the offset and the repetition frequencies, each require independent stabilization via modulators and feedback loops. The residual phase noise in the two feedback loops contributes directly to the stability of the OFC as a whole, and plays a vital role in several applications. For example, ultra-stable frequency combs (i.e., those exhibiting very low residual phase noise), are used to synchronize to optical clocks which have a high degree of stability compared to conventional atomic clocks based on radio frequency (RF) transitions and are thus being considered in the redefinition of the second [3, 4]. Ultra-stable frequency combs have also been used to generate low-noise microwave reference signals that can help to improve radar systems and telecommunication . The stability of these low phase noise applications is currently limited by that of the continuous wave (CW) laser used, which has dramatically improved in recent years and may soon approach 10−17 stability at 1 s . OFCs used to synchronize to these systems need to demonstrate the same level of stability or better.
The robustness of the OFC is also an important aspect to consider for user-friendly and long term operation. OFCs based on fiber lasers have shown to fulfill these requirements [7–9]. However, compared to OFCs based on bulk lasers, such as Ti:Sapphire frequency combs, fiber frequency combs are inherently more noisy at high frequencies due to their high-loss-high-gain cavity configuration. Therefore, ultra-stable fiber OFCs require broadband feedback modulators to compensate for high-frequency noise. When it comes to stabilizing the offset frequency, numerous methods have been implemented over the years, including modulation of the pump laser diode current, or use of acousto-optic  and graphene [11, 12] modulators. Graphene modulators especially, have a broadband feedback bandwidth with a cutoff frequency which reaches the MHz-level. Another approach to handle the offset frequency is to use difference frequency generation [13, 14], which leads to the offset frequency noise passively canceling out and care is focused on the stabilization of the repetition frequency alone. When it comes to the repetition frequency, fewer methods exist to manage it, and they typically involve controlling the cavity length via piezo-electric transducers (PZTs) and electro-optic modulators (EOMs). Although EOMs can be operated with a much broader bandwidth than PZTs, it is difficult to exceed the MHz-range. The feedback bandwidth, when using EOMs, is limited by the mechanical resonances triggered by piezo-electric effects [15, 16]. These resonances are inevitable since the electro-optic effects are accompanied by the piezo-electric effects due to asymmetrical charge distribution in electro-optic crystals. Since Hudson et al. first demonstrated the control of the repetition frequency with an EOM in 2005 , no devices have been proposed to overcome this issue.
In this paper, we propose a novel way to control the cavity length of a fiber based OFC via the magnetic field. The advantage of using the magnetic field is the suppression of mechanical resonances, thanks to the rather small magnetomechanical effects of non-magnetic dielectric materials, which would allow a broader feedback bandwidth. We demonstrated that the magneto-optic modulator (MOM) could work as a cavity length modulator by applying it to a Yb:fiber based OFC. To evaluate its performance, we stabilized one longitudinal mode of the OFC to a reference CW laser using the MOM, and a residual phase noise of sub-hundred mrad was achieved.
2. Magneto-optic modulator (MOM) concept and design
A magnetic-sensitive medium exhibits circular birefringence, i.e., difference in the refractive indices for left- and right-handed circular polarization, by applying a magnetic field. This is known as the Faraday effect. Thus, when two counter-rotating circularly polarized pulses propagate through a medium, as shown in Fig. 1, each pulse experiences a different propagation velocity. As a consequence, through modulation of the applied magnetic field, the effective optical path length of a circularly polarized light can be controlled.
The modulation depth is equal to 1/2π × VBlλ, where V is the Verdet constant that quantifies the magnetic sensitivity of a medium, B is the magnetic-flux density, l is the length of the medium traveled by the light, and λ is the wavelength of the light. The Verdet constant is typically small, and the magnetic-field strength is also restrained. In order to increase the modulation depth, we opted for an optical fiber as the magnetic-sensitive medium to increase the interaction length. It should be noted that the polarization state of the light in the fiber needs to be kept circular for it to act as a MOM.
For this reason, we employed a spun fiber as the MOM. A spun fiber has a bow-tie structure like that of conventional polarization-maintaining fibers, which rotates along the axial direction of the fiber and enables to maintain circular polarization . These fibers are commonly used for optical-current sensors  and are commercially available at low cost. Spun fibers are magnetic-sensitive media with Verdet constants of approximately 1 rad/T/m at λ = 1000 nm . Such fiber-based MOMs can be conveniently applied to fiber lasers and support all-fiber configurations as well.
The design of our MOM is shown in Fig. 2. To apply a sufficiently strong magnetic field, and correspondingly enhance the modulation depth, a current is applied to a toroidal coil made of a copper wire with 200 turns woven around and along the wound-spun fiber. The diameters of the toroid and the coils were approximately 70 mm and 2 mm respectively. The spun fiber (Thorlabs SHB1250) was 1 m in length and wound 5 times around the inside of the toroidal coils.
3. Experimental setup and results
A schematic diagram of the experimental setup is shown in Fig. 3. The MOM is integrated into a nonlinear polarization rotation mode-locked Yb:fiber laser. The repetition frequency was 68 MHz and the net dispersion was approximately set to zero. The modulation depth of the MOM was estimated to be a few mHz/A in terms of repetition frequency variation per unit current applied to the coil. A quarter-wave plate was placed between the isolator and the collimator to adjust the incident polarization of the pulses into the spun fiber. The slow drift of the cavity length was compensated by the PZT, onto which the cavity folding mirror was mounted. The slow drift was caused by thermal issues of the MOM. To suppress it, a simple temperature controller was implemented to the MOM.
First, we evaluated the MOM stroke in the optical region by measuring the optical beat note between the free-running mode-locked Yb:fiber laser and the ultra-stable CW laser with sinusoidal modulation current to the MOM. As mentioned before, the variation of the repetition frequency was so small (mHz level for 1-A modulation current) that it could not be measured using a frequency counter. In the optical region, however, the variation of the longitudinal modes was magnified by approximately 4 × 106 times assuming that the offset frequency was static. As shown in Fig. 4(a), a free-running optical heterodyne beat note between the Yb:fiber laser and the CW laser was measured by a spectrum analyzer. We emphasize that the Yb:fiber laser was not stabilized to avoid any unwanted distortion. A series of time slices of the beat spectrum, called a spectrogram, is shown in Fig. 4(b). One-Hertz sinusoidal current (0.8 A) was applied to the MOM, and the corresponding optical beat modulation was observed as indicated by a red curve. Figure 4(c) shows an FFT of the ridge of the spectrogram. An optical beat-modulation depth of 10 kHz at the 1-Hz modulation frequency was obtained. We note that the linear drift component was subtracted before the FFT to avoid a wrap-around artifact. Using MOM parameters including a Verdet constant, the calculated variation of the repetition frequency was approximately 3 mHz and the corresponding modulation depth in optical region was 12 kHz assuming the offset frequency was not varied. The result was in good agreement with the calculation value.
To evaluate the feedback performance of the MOM as a cavity length modulator, one of the longitudinal modes of the mode-locked laser was stabilized to an ultra-stable CW laser. Since the MOM modulation depth is rather small even in the optical region, the assistance of other modulators is required. Thus, the MOM feedback was not employed, but a combination of a fast-feedback loop (<100 kHz) to the current of the pump laser diode and a slow-feedback loop (<10 Hz) to the PZT was. This feedback scheme was evaluated via the RF spectrum of the stabilized beat-note. Secondly, the RF spectrum of the beat-note, which was stabilized by additionally feeding back to the MOM, was measured again. The differences between these two spectra, as shown in Fig. 5(a), indicate the effectiveness of the MOM. The spectrum with the blue line was achieved when modulating the current of the pump laser diode and the PZT, while the red curve was obtained when additionally using the MOM. The resulting phase noise spectra are shown in Fig. 5(b), and the residual phase noise integrated from 100 Hz to 3 MHz improved to 92 mrad when additionally using the MOM. It is worth noting that there are no noticeable spikes due to mechanical resonances in the spectra, which sometimes appear when using EOMs . The absence of such spikes supports the idea that the MOM is free from disruptive mechanical resonances.
To estimate the feedback bandwidth, the electronic circuit equivalent to the MOM, as shown in Fig. 6(a), was considered. The circuit consisted of an LR low-pass filter. The toroidal coil acted as the inductor (L) and had an inductance of 1.4 μH, whereas the resistor (R) had a 10-Ω resistance and converted voltages to currents. The value of R was determined by considering the current and voltage limits of the MOM driver (composed of high-speed, high-output-current operational amplifiers). The cutoff frequency was measured with a network analyzer and was found to be approximately 1 MHz, as shown in Fig. 6(b). This measured value was in good agreement with the calculated value of R/(2πL) = 1.1 MHz. As indicated by the equation of the cutoff frequency of a LR low-pass filter, to expand the feedback bandwidth of the MOM, the inductance of a toroidal coil should be as small as possible. Assuming the toroidal coil to be an ideal solenoid, the inductance is determined by the length and the diameter of the coil. A shorter coil has a smaller inductance and a shorter interaction length. These factors result in a broader bandwidth and a smaller modulation depth, respectively. In other words, varying the length of a coil leads to a tradeoff between the feedback bandwidth and the modulation depth. On the other hand, a smaller coil diameter leads to a reduced inductance (correspondingly, an increased feedback bandwidth) while the same modulation depth is maintained. The coiling diameter of the toroidal coil was 2 mm, a limitation set by winding by hand. Because the coil inductance is proportional to the square of the coil diameter, MOMs with smaller coiling diameters, such as industrially produced ones, would probably work well. Ultimately, the MOM bandwidth might be limited by the capacitance of the coil itself. However, this should not be an issue up to the tens of MHz. Therefore, the MOM has the potential to obtain much broader feedback bandwidths than conventional modulators such as EOMs.
Finally, we discuss the versatility of the MOM. In this paper we present a fiber-based MOM, however, bulk MOMs are also an option for precise and broad-bandwidth stabilization in bulk lasers, including OFCs with high repetition frequencies. Crystals of TGG and TSAG have Verdet constants that are 3- to 50 times larger than those of spun fibers , helping to save on interaction length. Here, the laser cavity followed a ring configuration, however a linear cavity would enhance the stroke, since a double-pass through a magneto-optic medium results in the rotation of the polarization, or the net-phase offset, being doubled. In this way, MOMs show much promise.
We developed a magneto-optic modulator for the precise control of the cavity length of an OFC. It was able to circumvent mechanical resonances which typically restrict the feedback bandwidth of conventional modulators. When using the MOM for the phase stabilization of a single longitudinal mode of the mode-locked laser to a CW laser, the integrated phase noise was successfully reduced down to 92 mrad from 128 mrad when phase-stabilization was carried out without it. We also discussed the current limitations of the MOM as a feedback element, as well as multiple points upon which to improve in the future. The MOM has the potential to achieve much broader feedback bandwidths than those achieved by the transducers currently used for the same purpose, such as EOMs, and it may greatly contribute to the next generation of ultra-stable frequency combs.
This research project was carried out in support of the Photon Frontier Network Program and Photon and Quantum Basic Research Coordinated Development Program of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
References and links
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