We investigate the frequency offset in the frequency stabilization of a semiconductor laser based on a frequency-dithering technique. An analytical model is presented to describe the effects of the amplitude modulation and the phase delay between the amplitude and frequency modulation on the frequency stabilization. We also experimentally and analytically show that the frequency offset could be reduced by using an appropriate phase-sensitive detection.
© 2007 Optical Society of America
Frequency-stabilized semiconductor lasers have many applications, such as lightwave communications, optical sensors, and high-resolution spectroscopy –. There have been many efforts expended in improving the frequency stability of semiconductor lasers by locking to the frequency references such as Fabry-Perot cavities  and atomic/molecular spectral lines –. Most of these stabilization techniques utilize frequency-modulated (i.e., frequency-dithered) lasers and heterodyne detection in order to derive the feedback signal proportional to the first derivative of the frequency reference . Basically, to modulate the optical frequency of a semiconductor laser, a small sinusoidal current is injected to the semiconductor laser. However, it is well-known that the output power of the laser is also modulated by the modulation current, which in turn, causes a frequency offset in the frequency stabilization . In applications that require absolute frequency accuracy, these frequency offsets can be significant and must be taken into account.
In this paper, we newly found out that this frequency offset depends on the phase delay between amplitude-modulated (AM) and frequency-modulated (FM) components of the laser output. Thus, to investigate the mechanism of the frequency offset, we explicitly calculate the effects of the AM component and the phase delay between AM and FM components on the frequency stabilization process. In addition, our experimental and analytical results show that the frequency offset could be reduced significantly by using a phase-sensitive detection technique.
Figure 1 shows the experimental setup used to measure the frequency offset caused by the AM component. The distributed feedback laser (DFB) laser is slightly dithered by using a small sinusoidal current generated from the oscillator. Thus, the optical power and frequency of DFB laser can be described as 
where P 0 is the average output power of the DFB laser, ΔP 0 is the peak amplitude of the sinusoidal modulation of the laser power, ω is the modulation angular frequency, ν 0 is the center frequency of the DFB laser, Δν 0 is the peak deviation of the laser frequency induced by the sinusoidal current, and ϕ is the phase delay between the amplitude and frequency modulation. In the experiment, the output signal of DFB laser was sent to the wavelength meter to measure the operating frequency (wavelength). However, a small portion was tapped and sent to the Fabry-Perot filter (i.e., frequency reference). Assuming the transmission characteristics of the Fabry-Perot filter is a slowly-varying function, the output signal of the Fabry-Perot filter, O(t), can be described as
where T(ν 0) and T′(ν 0) represent the transmission function of the Fabry-Perot filter and its first derivative, R is the reflectance of the mirrors, and FSR is the free spectral range, respectively. Then the output of the Fabry-Perot filter was sent to a photodetector that was connected to a lock-in amplifier. The lock-in amplifier consisted of a multiplier, a low-pass filter, and a phase delay module. Thus, the output signal of the photodetector is multiplied by the reference signal, and then sent to the low-pass filter. As a result, the output signal of the lock-in amplifier, S(t), can be expressed as
where θ is the phase delay of a reference signal in the lock-in amplifier. In Eq. (3), the first and second terms represent the effects of AM and FM components, respectively. To lock the laser frequency to the resonance peak of the Fabry-Perot filter, it should be necessary to set the feedback signal (S(t)) to zero. However, Eq. (3) indicates that the AM component of the DFB laser induces a frequency offset after the stabilization process (i.e., S(t)=0 for Δν 0≠ 0). From Eq. (3), the frequency offset, ν offset, can be analytically described as
It should be noted that the first term of Eq. (3) can be eliminated by setting θ to π/2. This means that, in this case, there is no frequency offset after the stabilization process. However, if φ is 0 or π, then the ν offset does not depend on θ in Eq. (4). Thus, in this last case, there is no way to remove the frequency offset. Hence one needs some phase delay (ϕ≠ 0,π) between amplitude and frequency modulation to be able removing the frequency offset in the stabilized laser.
3. Experimental Results
To investigate the frequency offset, we performed the experiment as shown in Fig. 1. The output power of a DFB laser was set to be 3 dBm. The temperature dependence of our DFB laser was measured to be about -11.7 GHz/°C. Thus, the temperature of the DFB laser was controlled to be less than ±0.005 °C by using a temperature controller. Figure 2 shows the frequency-deviation rate of our multiple-quantum well (MQW) DFB laser measured while varying the dithering frequency. In this experiment, the peak deviation of the laser frequency was directly measured at the RF spectrum analyzer after beating with a reference external-cavity tunable laser. The result shows that the frequency-deviation rate (GHz/mA) is decreased as the dithering frequency is increased in a low-frequency region (<100 kHz). This is mainly because the lasing frequency of semiconductor laser, in this spectral region, is modulated due to the temperature variation . Basically, the lasing frequency is modulated due to the thermal modulation of the active region of a DFB laser when the injection current is modulated. Thus, the frequency deviation caused by thermal modulation is decreased as the dithering frequency is increased since the thermal response of DFB laser is inversely proportional to the frequency. On the other hand, when the dithering frequency is higher than 100 kHz, the frequency-deviation rate is increased with the dithering frequency. This is mainly because the lasing frequency is modulated also due to the carrier-density modulation (i.e., frequency chirp) which is increasing with the modulation frequency.
To obtain the magnitude of the phase delay between the amplitude and frequency modulation, the output amplitude of the lock-in amplifier was measured while varying the laser frequency (by controlling the temperature of the DFB laser) resulting in the first-derivative profile. Experimental points, for which the lasing frequency was measured by the wavelength meter, were fitted with the calculated profile for varying phase delay as shown in Fig. 3. In this experiment, ΔP 0 and θ were adjusted to be 0.2 mW and π/2, respectively (at a modulation frequency of 10 kHz). The measured data agree well with the fitting curve. From this curve, φ was calculated to be -0.18π. In general, the magnitude of φ is increased with the dithering frequency. However, it should be noted that, when the dithering frequency is less than 1 kHz, the frequency offset could not be removed sufficiently. This is because φ is close to 0 in the low-frequency region . Thus, in our experiment, the dithering frequency was set to be 10 kHz. The frequency-dithered optical signal passes through the Fabry-Perot etalon filter. We have constructed Fabry-Perot etalon filters by coating both sides of 1.04-mm-thick polished fused-silica glasses with seven layers of TiO2/SiO2 (free spectral range = 100 GHz, reflectance = 0.3, 0.4, 0.5, finesse = 2.5, 3.3, 4.4). The temperature dependence of the resonance frequency of this quartz-based etalon is about -1 GHz/°C. The frequency stability of this filter was controlled to be better than ±50 MHz by using a temperature controller.
In the experiment, we measured the frequency offset while changing various types of Fabry-Perot etalon filters (reflectance = 30, 40, 50 %). Figure 4 shows the frequency offset measured while varying θ (by using a commercial digital lock-in amplifier), in comparison with theoretically calculated lines. As expected, the frequency offset increases as the reflectance of Fabry-Perot etalon filter decreases. The measured data agree well with the theoretically calculated curves obtained by using Eq. (4). The results show that the stabilized frequency is very sensitive to θ. For example, the frequency offset was measured to be about 10 GHz after the stabilization process when R and θ were 50 % and 0.46π, respectively. However, it should be noted that, this large frequency offset due to the AM component can be reduced significantly by setting θ to π/2. Thus, our results confirm that the frequency offset in the frequency stabilization of a semiconductor laser can be removed simply by adjusting the phase delay of the reference signal in the lock-in amplifier (i.e., phase-sensitive detection). However, in a practical system, this phase delay cannot be thoroughly controlled due to the instrumental noise. In this case, the frequency offset can be suppressed by using a narrow-linewidth frequency reference. Figure 5 shows the frequency offset calculated while varying the spectral linewidth of the frequency reference in high-resolution spectroscopy. The results show that the frequency offset can be reduced within 20 MHz when the spectral linewidth of the frequency reference is less than 3 GHz (in this case, we assumed that θ is controlled to be within 0.49π).
We have investigated the frequency offset, due to amplitude modulation, in the frequency stabilization of semiconductor lasers based on a frequency-dithering technique. To present an analytical model to describe the frequency offset, we introduced as a parameter the phase delay between the amplitude and frequency modulation, which enabled the reduction of the frequency offset by using a phase-sensitive detection. This model was confirmed by performing a frequency stabilization experiment using various Fabry-Perot etalon filters. The results showed that the frequency offset could be suppressed dramatically by adjusting the phase delay of the reference signal in the lock-in amplifier to π/2.
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