Narrow linewidth semiconductor laser with a multi-period-delayed feedback photonic circuit

Chenwei Zhang; Chenwei Zhang; Changda Xu; Changda Xu; Ya Jin; Ming Li; Wei Li; Yu Liu; Haiqing Yuan; Jinhua Bai; Junming An; Ninghua Zhu; Ninghua Zhu; Ninghua Zhu

doi:10.1364/OE.458327

1. Introduction

Since the narrow linewidth laser diodes (LDs) exhibit advantages of extreme low phase noise, long coherent length, compact size, low energy consumption, long lifetime and low cost, this kind of laser has found a wide range of applications, such as optical atomic clock [1,2], laser gyroscope [3,4], light detection and ranging [5,6], laser spectroscopy [7], optical frequency synthesis [8], microwave photonics [9,10], coherent optical communication [11], and so on. In comparison with the commercial available narrow linewidth solid state lasers and fiber lasers, the narrow linewidth laser diodes are more suitable for mass manufacturing, as well as are more popular in the design and fabrication of integrated photonic devices. However, the typical linewidth of LDs already put into mass manufacturing is mostly in the order of MHz [12].

To develop narrow linewidth laser diodes with better spectral purity, and consequently strong coherence, a great many of investigations were carried out with non-integrated lasers using bulk components. With bulk optical feedback, such as optical grating [13] and Bragg fibers [14,15], LDs with the linewidth of sub-kilohertz have been obtained. Using a fiber-based self-optical feedback, the linewidth of a wavelength tunable distributed feedback (DFB) LD array was reduced from several MHz to less than 8 kHz over the full C-band range [16]. Due to the restriction of size and mass and the lack of long-term stability, even miniaturized bulk lasers obviously does not satisfy the requirement of integrated photonic circuits [9]. For the requirement of applications, hybrid laser diodes, in which a semiconductor gain medium and a feedback photonic circuit (FPC) with long extended resonator length, low-loss and a highly frequency selective feedback was optically coupled, has been extensively investigated. Narrow linewidth can be achieved due to increased photon lifetime of the resonated laser photons that is associated with an extended cavity length [17]. By integrating a DFB laser array with a silica-based external optical feedback planar lightwave circuit, a hybrid laser with a narrow linewidth of less than 10 kHz was obtained [18]. With the advent of high-quality intra-cavity microring resonators (MRRs), new paradigms were introduced to get chip-based narrow linewidth hybrid diode lasers, making use of MRRs in the feedback circuit. Due to the multiple roundtrips in the MRR, this length extension contributes to a narrow laser linewidth [19]. R. Oldenbeuving et al. coupled an optical gain chip to an external cavity that incorporates a double MRRs, and a narrow spectral bandwidth of 25 kHz was offered [20]. Recently, M. A. Tran et al. demonstrated a Si/InP hybrid laser comprising a single gain section and a Vernier filter composed of three MMRs, and the linewidth was narrowed to be 220 Hz [21]. It should be noted that by using this kind of FPC, laser linewidth can be significantly reduced. However, the oscillating laser mode should be in resonance with the Vernier filter and the MRRs need to have a high finesse [22].

In this paper, we design a new type of FPC constructed by a Sagnac ring and two coupled rings, namely a multi-period-delayed FPC (MPDF), which can be utilized to realize significant laser linewidth reduction when the wavelength of LD is continuously tuned. Using the DFB-LD and MPDF, a narrow linewidth semiconductor laser is fabricated and the laser output characteristics are studied experimentally.

2. MPDF design and elaboration on the mechanism

The design and the optical microscope image of the MPDF are shown in Fig. 1 (a) and (b), respectively. The LD chip is a self-made butterfly packaging distributed feedback (DFB) LD with an effective cavity length of 900 µm. The coating deposited on the back facet (BF) of the LD chip is high reflection at 1.5 µm (R@1.5µm =90%), while that on the front facet (FF) is anti-reflection at 1.5 µm (R@1.5µm =1%). The packaged LD chip was controlled at 25°C by a temperature controller with a resolution of 0.005°C. Then the laser was coupled into the MPDF via an optical fiber. The MPDF was consisted with a Sagnac ring (ring 1) and two coupled rings (ring 2 and ring 3), as shown in the inset of Fig. 1(a). The lengths of ring 1, ring 2, and ring 3 are 42 mm, 31.5 mm and 34.5 mm, respectively.

Fig. 1. (a) Design of the MPDF. OC: Optical coupler; BF: Back facet, FF: Front facet. (b) Optical microscope image of the MPDF.

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The MPDF was fabricated on a 6-inch silica-on-wafer with the standard chip-manufacturing process. Firstly, the silicon substrate was thermally oxidized and the core-layer was deposited by plasma-enhanced chemical vapor deposition (PECVD). Then, to make the material more compact, the core-layer was accurately annealed above 1100 °C. Prior to fully etching the core-layer by inductively coupled plasma dry etching process, we carefully patterned the waveguides through ultraviolet lithography. Finally, the top-layer was deposited and annealed the same as the core layer [23]. The size of the packaged MPDF is 2 cm × 2 cm, hence the full view of the MPDF was obtained by an optical microscope, as shown in Fig. 1(b). To show the structure of the MPDF more clearly, insets (1-4) corresponding to four key local parts of the MPDF were also given. The refractive indexes of the SiO₂ cladding and the SiO₂ core are respectively 1.4448 and 1.4737 at 1550 nm by controlling the doping concentrations.

The major difference between MPDF and the Sagnac type feedback [16] is that the light returned back into the LD chip on the front facet was mainly originated from four kinds of processes for the LD with MPDF. The first was the direct reflection of the intracavity laser on the facet. The second was the light circulated once in the Sagnac ring experiencing a time delay of 2τ₀+τ₁, where τ_i = L_i/c and L_i are the transmit time in path i (i = 0, 1, 2, 3 indicate the circuit between the LD chip and Sagnac ring, the Sagnac ring, ring 2 and ring 3, respectively), c is the light speed. The third was the light went through the Sagnac ring once and ring 2 for n times, where n is arbitrary integer. The fourth was the light went through the Sagnac ring once, ring 2 for n times and ring 3 for m times, where m is also arbitrary integer. Neglecting the other processes, the laser can be feedback to the LD chip by a great many of delay periods, such as 2τ₀+τ₁, 2τ₀+τ₁+τ₂, 2τ₀+τ₁+2τ₂, …, 2τ₀+τ₁+ nτ₂, 2τ₀+τ₁+τ₂+τ₃, 2τ₀+τ₁+τ₂+2τ₃, …, 2τ₀+τ₁+τ₂+ mτ₃, 2τ₀+τ₁+ nτ₂+ mτ₃, …, etc. Since the laser photons emitted in different times can be feedback at the same time, the oscillation wavelength of the LD can be pulled back to the initial position immediately once an environmental disturbance was occurred, and the frequency stability relative to the wavelength jitter can be eliminated. Furthermore, with the help of the optical feedback introduced by the MPDF, the impact of the spontaneous emission to the laser longitudinal mode was also lowered significantly.

The major difference between MPDF and the previous reported microring resonators (MRRs) is the feedback spectral response of the laser passing through these feedback structures. The MPDF exhibited a low Q transmission spectra in the view of MRR theory, but MRRs exhibited a high Q reflection spectra [24]. Figure 2 schematically shows the experimental scheme for measuring spectral response of the MPDF, where the squares in the dashed box indicate the instruments used in this experiment. A light-wave with a wavelength of 1550 nm and a power of 10 dBm was emitted by a LD and injected into a Mach-Zehnder modulator (MZM). The optical signal was intensity-modulated by the radio frequency (RF) signal generated from a vector network analyzer (VNA) via the MZM. The intensity-modulated optical signal passing through the MPDF was detected by a high speed photodetector (PD). The photo-current from PD was sent back into the VNA and analyzed. Figure 3 shows the measured spectral response of the MPDF in a frequency range from 10 MHz to 40 GHz.

Fig. 2. Experimental scheme for measuring spectral response of the MPDF. MZM: Mach-Zehnder modulator; ISO: Isolator; PD: Photodetector; VNA: Vector network analyzer.

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Fig. 3. Measured spectral response of the MPDF.

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It is apparent that the effective frequency response of the MPDF is the reflection spectra of the combination of the Sagnac ring and the other two rings, consequently the linewidth reduction mechanism obeys the multi-period feedback principle in time domain. The coherent superposition of the feedback laser passing through different paths leads to the frequency response spectrum with periodical fluctuations, while the hollows corresponds to the period of 2τ₀+τ₁+ nτ₂+ mτ₃ with large values of m and n. Moreover, due to the low Q of MPDF and the dense distribution of the feedback periods, the phase of the feedback laser has low impact on the laser linewidth.

3. Theoretical modeling

Considering the difference between the operating schemes of DFB-based laser and Fabry-Perot laser, the linewidth behaviors of the DFB-based laser with MPDF were analyzed by making an approximation that the DFB-based laser was equivalent to a Fabry-Perot laser with varied threshold gain (g_th), effective resonator length (L_ec), total loss (δ_t) and mirror loss (δ_m).

According to the coupled wave theory of the DFB-based laser [25], using the transfer matrixes of the uniform gratings and the two facets of the LD, as well as the boundary condition derived from the self-excitation behavior of the laser, the threshold condition of the DFB-based laser can be deduced as:

(1)$$\left( {\sqrt {{R_1}{R_2}} {u_ + } + \sqrt {{R_2}} {v_ - } + \sqrt {{R_1}} {v_ + } + {u_ - }} \right)/\sqrt {({1 - {R_1}} )({1 - {R_2}} )} = 0,$$

where

(2)$$\begin{array}{l} {u_ \pm } = \cosh ({\gamma {L_c}} )\pm ({\alpha - i\delta } )/\gamma \times \sinh ({\gamma {L_c}} ),\\ {v_ \pm } ={\pm} i\kappa /\gamma \times \sinh ({\gamma {L_c}} ), \end{array}$$

R₁ and R₂ are the intensity reflectivity of the coatings on the LD-chip’s dual facets. L_c, Λ, n_eff and Δn are the grating length, period, effective refractive index and its variation of the uniform grating, respectively. g and α_g are the gain coefficient and intrinsic loss of the semiconductor, respectively. α = (g-α_g)/2 is the gain coefficient of the grating. λ₀ = 2Λn_eff is the Bragg wavelength, κ = πΔn/λ₀ is the coupling coefficient of the grating. δ = 2πn_eff(1/λ-1/λ₀) is the shift of the Bragg wavelength. γ = [κ²+(α-iδ)²]^1/2 is the transfer constant.

Using Eqs. (1) and (2), g can be numerically calculated with different value of λ, then the total loss and the excitation laser wavelength can be obtained by the relation that δ_t = g_th = g_min, where g_min is the minimum value of the numerical calculation of g.

Moreover, L_ec and δ_m can be calculated by [26,27]

(3)$${L_{ec}} = \tanh ({\kappa {L_c}} )/({2\kappa } ),$$

(4)$${\delta _m} ={-} \frac{\kappa }{{2\tanh ({\kappa {L_c}} )}}\ln \{{{{[{1 - \tanh ({\kappa {L_c}} )} ]}^2}({1 - {R_1}} )({1 - {R_2}} )} \}.$$

Based on the above discussion, an approximate estimation on the output behaviors of the three kinds of lasers (single LD, LD with Sagnac feedback, LD with MPDF) can be given. According to Henry’s and Fan’s theories [17,28], the linewidth of the laser without feedback (Δν) and that of the laser with MPDF (Δν_M) can be written as:

(5)$$\Delta \nu = \frac{{V_g^2h\nu {n_{sp}}{\delta _t}{\delta _m}({1 + {\alpha_H}^2} )}}{{8\pi {P_1}\left( {1 + \frac{{{r_1}}}{{{r_2}}}\frac{{1 - {R_2}}}{{1 - {R_1}}}} \right)}},$$

(6)$$\Delta {\nu _M}(\omega )= \frac{{V_g^2h\nu {n_{sp}}{\delta _{t,M}}(\omega ){\delta _{m,M}}(\omega )({1 + {\alpha_H}^2} )}}{{4\pi {F^2}(\omega ){P_1}\left( {1 + \frac{{{r_1}}}{{|{{r_o}(\omega )} |}}\frac{{1 - {r_o}{{(\omega )}^2}}}{{1 - {R_1}}}} \right)}},$$

where r_i (i = 1, 2) are the amplitude reflectivities of the coatings defined with the relation of R_i= r_i². V_g= c/n_eff is the group velocity inside the LD chip, δ_m,M is the mirror loss coefficient for the laser with MPDF that can be calculated using Eq.(4) with R₂ being replaced by r_o²(ω). δ_t,M is the total loss coefficient per roundtrip for the laser with the MPDF. δ_t,M can be calculated using Eqs. (1) and (2) with R₂ being replaced by r_f²(ω). r_o(ω) and t_o(ω) are equivalent amplitude reflectivity and transmittance at the light emission port (‘Output’ in Fig. 1) when the MPDF is employed (r_o²(ω)+t_o²(ω) 1), that can be expressed as:

(7)$$\begin{array}{l} {t_o}(\omega )= {t_x}(\omega )+ {t_x}(\omega )\frac{{\sqrt 2 \exp [{ - {\alpha_M}{L_2}} ]\exp [{i\omega {\tau_2}} ]\left\{ {2 + \sqrt 2 \exp [{ - {\alpha_M}{L_3}} ]\exp [{i\omega {\tau_3}} ]} \right\}}}{{4 - 2\sqrt 2 \exp [{ - {\alpha_M}{L_2}} ]\exp [{i\omega {\tau_2}} ]}}\\ + {t_x}(\omega )\frac{{\exp [{ - {\alpha_M}({{L_2} + {L_3}} )} ]\exp [{i\omega ({{\tau_2} + {\tau_3}} )} ]}}{{2 - \sqrt 2 \exp [{ - {\alpha_M}({{L_2} + {L_3}} )} ]\exp [{i\omega ({{\tau_2} + {\tau_3}} )} ]}},\\ {t_x}(\omega )= \sqrt {\Gamma /2} \ast {t_2}\sqrt {{T_{oc}}} \exp [{ - {\alpha_M}{L_1}} ]\exp [{i\omega ({{\tau_0} + {\tau_1}} )} ]. \end{array}$$

τ_i are the times of the photon passing through the fiber connecting the LD chip and the MPDF (τ₀) and the ring i (i = 1,2,3) that defined as τ_i = L_i/c. r_f(ω), that is effective amplitude reflectivity at the front facet of the LD chip when the MPDF is employed, can be expressed as:

(8)$$\begin{array}{l} {r_f}(\omega )= {r_2}(\omega )+ {r_{x2}}(\omega )+ {r_{x2}}(\omega )\frac{{\sqrt 2 \exp [{ - {\alpha_M}{L_2}} ]\exp [{i\omega {\tau_2}} ]\left\{ {2 + \sqrt 2 \exp [{ - {\alpha_M}{L_3}} ]\exp [{i\omega {\tau_3}} ]} \right\}}}{{4 - 2\sqrt 2 \exp [{ - {\alpha_M}{L_2}} ]\exp [{i\omega {\tau_2}} ]}}\\ + {r_{x2}}(\omega )\frac{{\exp [{ - {\alpha_M}({{L_2} + {L_3}} )} ]\exp [{i\omega ({{\tau_2} + {\tau_3}} )} ]}}{{2 - \sqrt 2 \exp [{ - {\alpha_M}({{L_2} + {L_3}} )} ]\exp [{i\omega ({{\tau_2} + {\tau_3}} )} ]}},\\ {r_{x2}}(\omega )= \sqrt {({1 - \Gamma } )/2} \ast {T_2}\sqrt {{T_{oc}}} \zeta \exp [{ - {\alpha_M}{L_1}} ]\exp [{i\omega ({2{\tau_0} + {\tau_1}} )} ]. \end{array}$$

F is the linewidth reduction factor that can be written as:

(9)$$F(\omega )= 1 + \frac{c}{{2{n_g}{L_{ec}}}}Re \left\{ {i\frac{d}{{d\omega }}\ln [{r_f}(\omega )]} \right\} - \frac{{{\alpha _H}c}}{{2{n_g}{L_{ec}}}}{\mathop{\rm Im}\nolimits} \left\{ {i\frac{d}{{d\omega }}\ln [{r_f}(\omega )]} \right\}.$$

The definitions of the symbols mentioned above and their typical values are given in Table 1.

Table 1. Definitions and typical values of the symbols

View Table

Using equations (1-9) and the parameter values given in Table 1, the dependences of Δν, Δν_M, as well as the linewidth of the LD with a feedback circuit composed with only a Sagnac ring (Δν_S) on emitted power from the back facet of the LD chip (P₁) were simulated and shown in Figs. 4(a) and 4(b). It can be seen that Δν, Δν_M, and Δν_S were all reduced with increasing P₁, while the linewidth reduction ratio between Δν_S and Δν (LRR_S) was fixed at 0.47‰, and that between Δν_M and Δν (LRR_M) was decreased for more than 2 times that down to 0.19‰. Obviously, at the same value of P₁, i.e. identical intracavity laser intensity, the linewidth reduction capability of our MPDF is significantly better than the self-optical feedback structure with only a Sagnac ring, and the absolute variation of the laser linewidth caused by a fluctuation of the intracavity laser intensity is much smaller leading to a better linewidth stability.

Fig. 4. Laser linewidth as a function of P₁ for LD (a), LD with MPDF and LD with only a Sagnac ring (b).

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4. Experimental results and discussion

The testing system is schematically shown in Fig. 5. The laser beam from the output port of the MPDF was divided into two beams by an optical coupler (OC) beam splitter. One part with 10% power was delivered to an optical spectrum analyzer (OSA, Model: AQ6370D, YOKOGAWA) for wavelength test, and the other part was sent to a delayed self-heterodyne (DSH) system for linewidth measurement [29]. In the DSH system, the laser was firstly divided into two branches by a 50/50 OC. One branch passed through an optical delay line composed of 50 km optical fiber and a polarization controller (PC), and the other branch passed through an acoustooptic modulator (AOM) driven by the 80 MHz sinusoidal signal. Subsequently, two branches were combined by the second 50/50 OC. The ideal measurement resolution of the DSH system was 2 kHz according to theoretical prediction. Finally, the heterodyne signal from a photodetector was measured by a phase noise analyzer (PNA, Model: FSWP (1 MHz-50 GHz), ROHDE&SCHWARZ).

Fig. 5. Testing system. OC: Optical coupler; SMF: Single mode fiber; PC: Polarization controller; AOM: Acoustooptic modulator; PD: Photodetector; OSA: Optical spectrum analyzer; PNA: Phase noise analyzer.

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When the driving current of the DFB-LD was maintained at 100 mA, the laser wavelength and the frequency spectra of the DFB-LD and those of the DFB-LD with MPDF are measured using the testing system as shown in Fig. 6 and Fig. 7. The blue curve in Fig. 6 shows the optical spectrum of the DFB-LD. It can be seen that the central wavelength was 1550.86 nm with a side mode rejection ratio of 56 dB. Despite that the single-mode rate of DFB lasers (uniform gratings with two facets, one facet is high reflection and the other is anti-reflection) is usually limited to 70%, which is mainly related to its manufacturing process, single mode laser operation at a wavelength located within the reflection spectral region of the grating of the DFB LD was realized here at a driving current far beyond the threshold. For the DFB-LD with MPDF, both the central wavelength and the operating mode remained unchanged except that only the power at the central wavelength was lowered by 5 dB, as shown by the red curve in Fig. 6. Figure 7 displays the power spectra of self-beating signal of the DFB-LD with MPDF and that of the DFB-LD (in inset), respectively. In comparison with the 1.25 MHz linewidth DFB-LD, the linewidth of the DFB-LD with MPDF is lowered by three orders of magnitude, reaching 2 kHz.

Fig. 6. Optical spectra of the DFB-LD and the DFB-LD with MPDF.

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Fig. 7. Power spectra of self-beat signal of the DFB-LD with MPDF and that of the DFB-LD in the inset. RBW: Resolution bandwidth; VBW: Video bandwidth.

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Figure 8(a) is the linewidth evolution of the DFB-LD with MPDF when the LD chip temperature was tuned from 10 °C to 45 °C with a step of 1 °C at a fixed current of 100 mA. It can be seen, due to the LD chip temperature variation, the laser wavelength was correspondingly tuned from 1549.476 nm to 1552.658 nm, and the laser linewidth kept almost the same. To make an intuitive display, the relationship between the linewidth of DFB-LD with MPDF and the laser wavelength is shown in Fig. 8(b). It can be seen, even when the laser wavelength was tuned for more than 3 nm, which was limited by the wavelength tuning range of the DFB-LD, the laser linewidth was always reduced by three orders of magnitude, and located in a range from 1.5 kHz to 2.8 kHz. In other words, owing to our MPDF structure, the laser in a wide spectral region could transmit from the FPC freely and the linewidth reduction capability could be maintained simultaneously. This phenomena can be understood that due to the great linewidth reduction capability of the MPDF, even if the LD temperature was changed for tuning the laser wavelength, the absolute linewidth variation of the DFB-LD with MPDF caused by temperature change was so little that was comparable with or even less than the measurement error.

Fig. 8. (a) Wavelength and linewidth of the DFB-LD with MPDF when the LD chip temperature was tuned from 10 °C to 45 °C. (b) Linewidth of the DFB-LD with MPDF as a function of the laser wavelength.

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Beside the intrinsic linewidth, the frequency fluctuation is also a key parameter for narrow linewidth LDs. Using a frequency stabilized laser (NKT, Model: E15) with a wavelength of 1550.124 nm and the Allen deviation of 1.1788×10⁻¹⁵ @100 min acting as frequency standard, the frequency fluctuations of the DFB-LD and the DFB-LD with MPDF were measured by beat frequency method. Figure 9 shows the beat frequency of the DFB-LD and the DFB-LD with MPDF in 60-minutes measurement with a sampling period of 10 ms, respectively. It can be deduced that the Allen deviation of the DFB-LD was 3.9306×10⁻⁹@60 min, and that of the DFB-LD with MPDF was 5.0640×10⁻¹⁰@60 min. Note that the oscillation frequency stability of the LD operating in steady state measured by beat frequency method was mainly originated from the wavelength jitter and the irregularities in the spontaneous emission. Owing to the special structure of the MPDF, laser photons emitted in different times can be feedback at the same time, so that the oscillation wavelength of the LD can be pulled back to the initial position immediately once an environmental disturbance was occurred. Since the excitation laser wavelength was locked at a relative stable state, the frequency stability relative to the wavelength jitter can be eliminated. Furthermore, with the help of the optical feedback introduced by the MPDF, the impact of the spontaneous emission to the laser longitudinal mode was lowered significantly. Hence the frequency stability relative to the spontaneous emission was also reduced. Consequently, the DFB-LD with MPDF structure not only brings about a narrower laser linewidth, but also leads to a better laser frequency stability.

Fig. 9. Frequency fluctuations of DFB-LD and DFB-LD with MPDF.

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5. Conclusion

We proposed a new method named as MPDF. By coupling a DFB-LD with the MPDF, the reduction of the laser linewidth can be realized throughout when the wavelength of DFB-LD was tuned. Using the fabricated DFB-LD and MPDF, stable narrow linewidth laser operation was realized experimentally. When the DFB-LD was driven by 100 mA current and controlled at 25 °C, the DFB-LD with MPDF exhibited a linewidth of 2 kHz, which was 625 times lower than the DFB-LD without MPDF. More importantly, our MPDF provided special advantages that the linewidth reduction capability could be maintained when the wavelength of the DFB-LD was tuned in a range wider than 3 nm, which was limited by the wavelength tuning range of the DFB-LD. Meanwhile, the laser frequency stability could also be improved using the proposed technique, and the frequency fluctuation was reduced for nearly 8 times in comparison with the DFB-LD without MPDF. This kind of DFB-LD with MPDF can be easily embedded into the present integrated photonic devices based on LDs to improve the performance of the devices. Further improvement of the DFB-LD with MPDF can be expected when the parameters of the MPDF is optimized, for instance the splitting ratio of the Sagnac ring and the lengths of the rings.

Funding

National Natural Science Foundation of China (61835010).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but can be obtained from the authors upon reasonable request.

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Symbol	Physical meaning	Typical value
c	Light speed in vacuum	3×10⁸ m/s
λ₀	Bragg wavelength	1550.86 nm
α_H	Linewidth enhancement factor	5.4
n_sp	Spontaneous emission coefficient	2
n_g	group refractive index of the semiconductor chip	3.989
n_eff	Effective refractive index of the grating	3.2
L_c	Length of the semiconductor chip	900 µm
α_g	Intrinsic loss of the semiconductor	2507 m^-1
κ	Coupling coefficient of the grating	3000 m^-1
R₁	Intensity reflectivity of the LD chip’s back facet	0.9
R₂	Intensity reflectivity of the LD chip’s front facet	0.01
T_oc	Coupling efficiency of laser from the LD chip to MPDF	0.98
n_M	Refractive index of MPDF	1.4448
Г	Splitting ratio of the Sagnac ring	0.9
L₀	Length between the LD chip and MPDF	55 mm
L₁	Length of the Sagnac ring in MPDF	42 mm
L₂	Length of the ring 2 in MPDF	31.5mm
L₃	Length of the ring 3 in MPDF	34.5 mm
α_M	Absorption coefficient of the MPDF at the laser wavelength	15.8 m^-1
ζ	Backward coupling efficiency (Coupling efficiency of laser from MPDF back to the LD chip)	0.4

Narrow linewidth semiconductor laser with a multi-period-delayed feedback photonic circuit

Abstract

1. Introduction

2. MPDF design and elaboration on the mechanism

3. Theoretical modeling

4. Experimental results and discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (1)

Equations (9)

Optics Express