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Abstract
The coherent optical negative feedback scheme is systematically investigated by calculating rate equations that model a noise-added semiconductor laser coupled to a Fabry-Perot optical filter for the FM noise reduction. The calculated results indicate that the FM noise is minimized when a lasing frequency of the free-running laser matches a valley frequency of the filter (the point where power reflectivity becomes zero) under a specific feedback phase, where the slope of the electric field reflectivity for the lasing light and frequency discrimination efficiency to electric field amplitude of the feedback light becomes maximum. And the linewidth is also minimized at a lasing frequency corresponding to the valley frequency of the Fabry-Perot optical filter. It is also made clear that the laser frequency becomes less sensitive to the fluctuation of the injection current of the laser under optical negative feedback.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High coherence laser sources have attracted attention in many fields of applications, for example, digital coherent communications [1, 2], light detection and ranging [3], and multi-heterodyne spectroscopy [4]. Narrow linewidth semiconductor lasers are advantageous for use in these applications in terms of power consumption, size, and cost. However, semiconductor lasers generally suffer from a broad linewidth caused by a relatively low quality (Q) factor of the laser cavity. Distributed feedback (DFB) lasers with a lengthened cavity structure [5, 6], external cavity laser diodes (ECLs) with self-injection locking [7–10], and a resonant optical feedback scheme [11–13] had been investigated for obtaining narrow linewidth semiconductor lasers. For instance, linewidths of 88 and 70 kHz were achieved by using a 1500-μm-long DFB laser [6] and an ECL [8]. The linewidths achieved by using these kinds of laser structures were limited to several tens of kilohertz when a total cavity length was less than a few millimeters suitable for manufacturing a small chip. The resonant optical feedback scheme used a narrow-pass-band optical filter such as an ultra-high Q whispering-gallery-mode resonator for a self-injection locking, which obtained a linewidth of 200 Hz [11]. As another approach, a configuration of a semiconductor optical amplifier with tunable silicon ring filters as a laser cavity was investigated, and linewidths of several tens of kilohertz were reported [14–16].
The resonant optical feedback uses DC feedback lights for the phase pulling effect which is the so-called self-injection locking, and the high-Q optical filter acts as a part of laser cavity. This induces unintended increase of FM noise due to the external cavity mode at specific frequencies depending on the external cavity length [17], and degrades performances such as bit error rate in the digital coherent communication. On the other hand, the coherent optical negative feedback scheme we proposed exploits a frequency discrimination of an optical filter set outside the laser cavity for a negative feedback control of a fluctuating lasing frequency and linewidth reduction to 3-kHz was demonstrated [18–20]. The optical negative feedback scheme does not rely on the external cavity mode unlike the self-injection locking. The most efficient reduction of FM noise is obtained when a lasing frequency of free running laser matches to a valley frequency of the FP optical filter (the point where power reflectivity becomes zero) under appropriate feedback light phase, in other words, a DC component of feedback light is essentially unnecessary and thus external cavity modes are removed in the optimal condition. From this aspect, the linewidth reduction based on the electrical negative feedback [21, 22] is significantly related to our approach. The lasing frequency is stabilized by the injection current in the electrical negative feedback, while it is stabilized by the feedback light in the optical negative feedback. The bandwidth of FM noise reduction under the electrical negative feedback, however, is limited by that of electrical circuits, and the system size becomes large. The optical negative feedback pushes the upper limit of the bandwidth up to several tens of gigahertz with a small size [18]. In this paper, we investigate a strategy of the coherent optical negative feedback scheme with attention to the stability of the lasing frequency by using rate equations. An optimum condition of frequency difference between a free-running semiconductor laser and an optical filter for obtaining the narrowest linewidth is clarified. This condition is also verified by experiments.
2. Principle of coherent optical negative feedback
The schematic of the coherent optical negative feedback scheme is shown in Fig. 1. The output light of a single-mode laser diode (LD) is reflected by an optical filter that acts as an optical frequency discriminator and is reinjected to the LD. Although the Fabry–Perot (FP) etalon is assumed as the optical filter in this study, other kinds of filters such as ring filters can be applied as well. The optical filter converts the FM noise in a lasing frequency of the LD into the electric field amplitude modulation of reflected light that modulates the lasing frequency of the LD negatively to reduce a frequency fluctuation. A process of the negative feedback is as follows.
- a. Lasing frequency shifts to higher (lower) side because of certain causes
- b. Reflectivity of the optical filter increases (decreases)
- c. Electric field amplitude of reflected feedback light increases (decreases)
- d. Photon density in the laser cavity increases (decreases)
- e. Carrier density in the laser cavity decreases (increases) through spontaneous emission
- f. Refractive index of the laser cavity increases (decreases) through carrier plasma effect
- g. Lasing frequency shifts to lower (higher) side
As a result, the lasing frequency of the LD is stabilized. The operation point shown in Fig. 1 is determined by a relative frequency difference between the free-running LD and optical filter, and it controls the strength and phase of the feedback light. The feedback phase is affected by a spatial separation between the LD and optical filter. These contributions are systematically investigated in the next section.
3. Numerical analysis of coherent optical negative feedback
3.1 Simulation model
In the previous studies [18–20], the FM noise and spectral linewidth reduction ratio of the optical negative feedback system were discussed by using transfer functions. This analysis, however, was approximated by the steady state of LD and could not describe a dynamic behavior of the lasing frequency when changing drive conditions of the LD. To evaluate detailed characteristics of the LD under the coherent optical negative feedback, we use the following rate equations [23],
where a single-mode DFB-LD coupled to a FP optical filter is assumed. The N(t) and represent the carrier density and slowly-varying amplitude of electric field, respectively. The feedback light term T represented by Eq. (3) considers reflected lights from the FP optical filter with a constant feedback phase ϕLoop. Here, a free spectral range (FSR) of 25 GHz and the full-width at half-maximum (FWHM) of 160 MHz in the power reflectivity spectrum are assumed for the filter. The number of round trips in the FP optical filter p is set to 200, which is large enough to describe assumed values of the FWHM and the extinction ratio. To introduce the electric field amplitude and phase fluctuations in lasing mode of the LD, the Langevin force F having a noise frequency fm and amplitude A defined as Eq. (4) is added to Eq. (2) [24]. In the analysis, a value of A is set so that the spectral linewidth of the LD becomes 10 MHz under a free running condition. The definitions of other parameters are listed in Table 1, and parameter values for the DFB-LD are taken from [25]. These rate equations are solved by the fourth-order Runge–Kutta method, and the power spectral density of FM noise (FMN-PSD) is calculated from a lasing frequency fluctuation [26].
Table 1. Parameters Used in Calculation
3.2 Frequency discrimination characteristic of optical filter
Typical characteristics of the FP optical filter are discussed to clarify a physical picture of the optical frequency discrimination. We define the differential frequency discrimination factor (feedback gain of optical filter) at DC as follows.
Here v denotes the amount of frequency fluctuation. The dependence of HOF|DC on ϕLoop and the detuning of the input light frequency from the valley frequency of the FP optical filter Δf and vertical cross-sections of HOF|DC are shown in Figs. 2(a) and 2(b). Positive and negative values of HOF|DC induce, respectively, negative and positive feedbacks for a fluctuation of the lasing frequency of the DFB-LD. Therefore, it is found that a condition of ϕLoop = −π/2 and a lasing frequency consistent with the valley frequency of the FP optical filter (Δf = 0), where the DC component of the feedback light is zero are most suitable for an efficient reduction of the FM noise. The separation between the filter and the laser induces a feedback loop delay which puts an upper limit of the noise reduction bandwidth. Therefore, the shorter separation is desirable.
Fig. 2 (a) Dependence of HOF|DC on ϕLoop and Δf. Valley frequency of FP optical filter is indicated by black dashed line. (b) Real part of field reflectivity of FP optical filter (solid black lines) and vertical cross-sections of HOF|DC (dashed black lines) under ϕLoop of −π/2, 0, π/2, and π. Red dashed lines indicate the negative feedback region.
3.3 Lasing frequency stability
Figure 3(a) shows the calculated results of the lasing frequency under the feedback condition in relation to that under the free-running condition for different specific ϕLoop values. These characteristics are obtained by increasing and decreasing f0. The horizontal and vertical axes are, respectively, detuning frequencies of the DFB-LD under the free-running condition ΔfFR and the feedback condition ΔfFB where each axis is normalized by a valley frequency of the FP optical filter. We confirm hysteresis characteristics where positive and negative sweeps of ΔfFR are indicated by red and blue curves, respectively. When ΔfFR reaches the positive feedback frequency range, the lasing frequency drifts into the next stable frequency. The observed hysteresis behaviors are attributed to an accelerated drift of the lasing frequency under positive feedback. When ϕLoop = −π/2, the negative feedback condition appears across both slopes of the power reflectivity of the FP optical filter within −6 GHz < ΔfFR < 7.5 GHz, and a sadden lasing frequency drift occurs at both ends. In the case of ϕLoop = −π/2, the sign of the Re[ln(T)] changes in accordance with the sign of Δf as shown in Fig. 2(b). So the sign of the slight DC component of reflected feedback light changes and affects in a different manner for [ln(T)] in Eq. (2) in accordance whether the lasing frequency is lower or higher than the valley frequency of the optical filter’s power reflectivity. It may become the cause of the asymmetric property of the lasing frequency against ΔfFR mentioned above. These results indicate that the coherent optical negative feedback scheme has a fine stability and controllability of ΔfFB and a tolerance for ΔfFR caused by such condition’s disturbances as thermal and electrical noises. Other typical conditions where ϕLoop = 0 and π lead to asymmetric hysteresis behaviors and a degraded amount of the reduction ratio of FMN-PSD in the negative feedback range. The condition of ϕLoop = π/2 is hard to be used for the negative feedback because of the narrow negative feedback operation condition. The ΔfFB dependence of the reduction ratio of FMN-PSD under ϕLoop = −π/2 and the corresponding transmittance of the FP optical filter are shown in Fig. 3(b). The reduction ratio of FMN-PSD reaches −41 dB at the valley frequency of the filter and decreases with detuning the ΔfFB value from the valley frequency. The enlarged view of the relation between ΔfFR and ΔfFB under ϕLoop = −π/2 and the corresponding reduction ratio of FMN-PSD are shown in Fig. 3(c). The non-linear relation between ΔfFR and ΔfFB observed at both ends comes from a limit of the negative feedback. This result is consistent with a relation between ΔfFB/ΔfFR and the reduction ratio of FMN-PSD SFB/SFR represented as follows.
Fig. 3 (a) Relation between ΔfFR and ΔfFB under ϕLoop of −π/2, 0, π/2, and π with transmission spectrum of FP optical filter (right side). Blue-painted areas indicate negative feedback range. Positive and negative sweeps of ΔfFR are indicated by red and blue curves. (b) ΔfFB dependence of reduction ratio of FMN-PSD under ϕLoop = −π/2 and corresponding transmittance of FP optical filter. Reduced ratio of FMN-PSD is defined as 10 × log10(SFB/SFR). (c) Enlarged view of relation between ΔfFR and ΔfFB under ϕLoop = −π/2 and corresponding reduction ratio of FMN-PSD. Dashed curve shows ΔfFR versus (ΔfFB/ΔfFR)2.
The dependence of the reduction ratio of FMN-PSD, SFB/SFR, on ϕLoop and ΔfFB is shown in Fig. 4. The dark region indicates that any stable lasing frequency does not exist because of the positive feedback, and the lasing mode shifts to the nearest stable negative feedback condition. When −π < ϕLoop < 0, a stable lasing frequency exists near the valley frequency of the FP optical filter, and a condition of maximum FM noise reduction can be found by sweeping ΔfFB. When ϕLoop ~π/2, a lasing frequency does not exist near the valley frequency of the FP optical filter where HOF|DC has a negative value. The maximum FM noise reduction is obtained when ϕLoop = −π/2, and the lasing frequency is favorably drawn near the valley frequency of the filter in this condition. Since an optimization of the FM noise reduction is done by sweeping the lasing frequency to match the valley frequency of the filter, the resulting lasing frequency can be precisely matched with the optimal condition. It means that the lasing optical frequency of the laser under the optical negative feedback can be precisely tuned around the valley frequency of the optical filter when slight increase of the FM noise and the spectral linewidth is allowed.
Fig. 4 Dependence of SFB/SFR on ϕLoop and ΔfFB. Valley frequency of FP optical filter is indicated by white dashed line. Within 3-dB down, region of reduction ratio of FMN-PSD from maximum reduction is indicated by black dashed curve.
4. Experiment
As an experimental support, we first confirm the typical characteristics of spectral linewidth of lasing mode and spectra of FMN-PSD under coherent optical negative feedback. Then we discuss the stability of the lasing frequency and the accompanying variation of the linewidth reduction. The experimental setup is shown in Fig. 5. The InGaAsP DFB laser is operated at a wavelength of 1532 nm with a current of 90 mA (threshold current is 9 mA). The spectral linewidth of the DFB laser under free-running condition is 11.5 MHz. The output light emitted from a laser facet with an anti-reflection (AR) coat and collimated by a rod lens is used for the coherent optical negative feedback and is also used for measurements of the linewidth and spectrum of FMN-PSD. The thickness, facet power reflectivity, and refractive index of the FP optical filter are 4 mm, 0.97, and 1.5 (glass) whose values are the same as the parameters used in the numerical analysis. The spectral linewidth is measured by the self-delayed heterodyne method [27]. A 40-km delay fiber leads to a resolution of 2.5 kHz. The lasing frequency shift is measured by the FP interferometer with a FSR and sweep time of 3.3 GHz and 50 ms. The digital oscilloscope has a sample rate of 5 Gsample/s, and the frequency resolution is 6 MHz. The detail of our linewidth reduction measurement had been reported in [20].
The measured relation between ΔfFR and ΔfFB under the optical negative feedback and its enlarged view are shown in Figs. 6(a) and 6(b). ΔfFR is shifted by tuning the injection current at a rate of 1.5 GHz/mA. The feedback enables a fine tuning of ΔfFB at a rate of 72 MHz/mA. This −13.2 dB reduction of the frequency shift by the injected current under feedback agrees with the square root value of the reduction ratio of FMN-PSD and hence that of the linewidth reduction ratio (−27 dB) according to Eq. (6), and we confirm its consistency with the theory. As shown in Fig. 6(b), nonlinear frequency change can be confirmed on left edge of the plots. In this experiment, since ΔfFR is swept from positive to negative by increasing the injection current, the nonlinear characteristic is not found on right edge of the plots where the lasing frequency is after the drift. The measured reduction ratio of FMN-PSD and transmitted power is shown in Figs. 6(c) and 6(d). In Fig. 6(c), the measured transmitted power indicates an extremal characteristic. The fitted transmission spectrum shows an optical filter that has a FWHM of 750 MHz and a finesse of 33.3. The finesse lower than the ideal value of 96 is attributed to slight misalignment for the FP optical filter. We use the misaligned condition to vary the lasing frequency in a wide range because the frequency resolution of our system is limited to 6 MHz. Therefore, the available reduction ratio of FMN-PSD (−27 dB), in this case, is inferior to the calculated one (−41 dB). The measured reduction ratio of FMN-PSD maximized at the valley frequency of the FP optical filter corresponds to the calculated one under ϕLoop = −π/2. The measured result was fitted by the calculated one indicated as the dashed black curve, where the finesse of FP optical filter, relaxation oscillation frequency of DFB laser, and coupling coefficient of feedback light were set to 33.3, 13 GHz, and 0.088, respectively. The simulation under ϕLoop = −0.502 π agrees well with the measured result. This fact demonstrates that narrow linewidth semiconductor lasers based on the coherent optical negative feedback scheme can operate at the vicinity of a valley frequency of the FP optical filter. Additionally, we measured the characteristics with ϕLoop ~π condition as shown in Fig. 6(d). The measured reduction ratio of FMN-PSD maximized at the slope of reflectivity that corresponds to the calculated result with ϕLoop of 1.11 π, finesse of FP optical filter of 56.0, and coupling coefficient of feedback light of 0.068. Through the measurements, we confirmed that the negative feedback range depends on ϕLoop in accordance with the calculated results. The values of ϕLoop were estimated by comparing the measured dependence of the reduction ratio of FMN-PSD on ϕLoop and ΔfFB with the calculated one. The experimental condition of ϕLoop = + π/2 could not be tested because the negative feedback gain is much lower than other conditions and an optimal condition exists in a frequency range where the FP optical filter has a high field reflectivity. Although ϕLoop is tuned and fixed by mechanical alignment of the DFB-LD, rod lens, and FP optical filter in this experiment, use of a piezoelectric element is expected to be feasible for this purpose. The thermal or electrical tuning of ϕLoop will be available in future monolithic configurations with an enhanced tolerance of mechanical vibration.
Fig. 6 (a) Measured relation between ΔfFR and ΔfFB under optical negative feedback (red circles). Dashed black line is guide for eye. (b) Enlarged view of (a). Solid black line is fitted line with least squares method. Measured reduction ratio of FMN-PSD (black circles) and transmitted power from FP optical filter (red circles) under (c) ϕLoop = −0.502 π and (d) ϕLoop = 1.11 π. Dashed red curves indicate fitted transmission spectrums of FP optical filter with finesse of 33.3 and 56.0. Dashed black curves indicate simulated results.
The measured optical spectrum under optical negative feedback (spectral linewidth is reduced to 3 kHz) is shown in Fig. 7. The very weak side-bands can be seen at 19, 25, and 50 GHz apart from the lasing peak. The side-band at 19 GHz is estimated to arise from the positive feedback predicted by the numerical analysis. However the intensity of the noise is very small, which indicates that the optical negative feedback system has an enough phase margin and operates stably. In case of small phase margin, the noise intensity due to positive feedback becomes large as reported in [19, 20]. The side-bands with 25 and 50 GHz are the transmission spectra of laser’s spontaneous emissions through the FP etalon because the output power through FP etalon was measured. Considering the feedback loop length of 3.5 mm, it is expected that the external cavity mode appears at 42.8 GHz. However, it was not observed in the optical spectrum, and we can confirm that the optical negative feedback is not accompanied by the external cavity mode.
Fig. 7 Measured optical spectrum under optical negative feedback. Spectrum is normalized by peak power. Resolution of optical spectrum analyzer is 0.01 nm.
5. Conclusion
We investigated the FM noise reduction and lasing frequency stability of a DFB-LD under the coherent optical negative feedback with a FP optical filter. The numerical analysis indicates that the negative feedback condition in which the reduction ratio of FMN-PSD becomes maximum can be set at the vicinity of the valley frequency of the FP optical filter when ϕLoop = −π/2. Under this condition, the stability of the lasing frequency of LD becomes high, and the operation point can be precisely tuned to the valley frequency of the optical filter owing to the reduced lasing frequency shift. In other words, it also indicates that the coherent optical negative feedback scheme has a high tolerance for disturbances like thermal noise or power supply noise. As an experimental support, we carried out the frequency tuning of a DFB-LD under the coherent optical negative feedback and measured a spectral linewidth and spectrum of FMN-PSD. We confirmed that the amount of lasing frequency shift by an injection current change is reduced by 13.2 dB under the negative feedback where the reduction ratio of FMN-PSD is measured to be −27 dB, which corresponds to the theoretical relation in Eq. (6). The FMN-PSD and hence the spectral linewidth are most reduced at the valley frequency of the FP optical filter in the case of ϕLoop = −π/2.
Acknowledgments
We acknowledge the stimulating discussion in the meeting of the Cooperative Research Project of the Research Institute of Electrical Communication, Tohoku University.
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