Abstract

A compact external cavity tunable laser based on a silicon hybrid micro-ring resonator is demonstrated. A theoretical model is also employed for design and analysis of the wavelength tuning performance of the device. In this model, the gain section of the device is simulated by a conventional multimode rate equation model, whereas all rest passive sections are modeled by the frequency domain method. Experimental results have shown that the output power of this device can reach 29 mW, with a linewidth less than 150 kHz. The tuning range is more than 17 nm in C-band with 60 dB side-mode-suppression-ratio (SMSR). This device shows a comparable performance with the commercial narrow linewidth laser as the source in coherent transmission systems.

© 2016 Optical Society of America

1. Introduction

In recent years, coherent optical transmission is proposed as a solution to increase the spectral efficiency [1]. A wavelength tunable laser with narrow linewidth is a key component in such coherent systems. Different types of tunable lasers have been presented. Y. Sasahata,et al. have reported a tunable DFB laser array composing 16-DFB lasers [2]. A.Sivananthan,et al. have demonstrated a tunable SG-DBR laser integrated with an asymmetric Mach-Zehnder interferometer (AMZI) [3]. Considering the cost and footprint requirement, people also introduced silicon photonics to fabricate the tunable lasers through the efficient coupling of light between the silicon on insulator (SOI) optical external cavity and the gain media [4–12]. However, either bonding the III/V material to SOI or hybrid integration [12] needs to introduce III/V material in CMOS processing line, which is not supported in most foundries. Besides, a long cavity is needed to achieve the required narrow linewidth in tunable lasers, which inevitably brings in an additional propagation loss to the optical loop. Considering the typical propagation loss in silicon waveguide is around 2.4 dB/cm, its negative effect can be quite significant. To obtain a narrow linewidth with low loss, micro lens coupling between the III/V gain and silicon photonics chip is a more realistic approach for its high maturity and large alignment tolerance.

In this paper, a low cost, low power consumption silicon narrow linewidth tunable laser based on a single micro-ring resonator (MRR) is demonstrated. This laser uses a lens coupled structure with a long cavity which is advantageous to linewidth narrowing. A hybrid model that combines the multimode rate equation approach and the frequency domain scattering matrix method is employed for device design optimization. This laser is housed in a butterfly package compatible with standard ITU-T grids. A linewidth as narrow as less than 150kHz has been obtained. The output power of the tunable laser is shown to be 29 mW. The tuning range is 17 nm in C-band with 60 dB side-mode-suppression-ratio (SMSR). The performance of this tunable laser is further evaluated in 4-QAM and 16-QAM coherent transmission systems.

2. Design of the device

2.1 Configuration of the device

The schematic layout and the photograph of the proposed laser is shown in Fig. 1. The laser cavity consists of a single silicon micro-ring chip, a gain chip with 1000 μm length, and an etalon. Compared with multi-microring tunable laser, we use only one micro-ring to obtain the narrow linewidth tunable laser without complex control. The schematic diagram of the silicon micro-ring chip is shown in Fig. 2(a), which consists of a micro-ring resonator (MRR), a 3 dB splitter/combiner, grating coupler (GC) and a spot size converter (SSC). The photograph of the GC, MRR and SSC are shown in Figs. 2(b)-2(d). This silicon micro-ring chip is fabricated on a silicon-on-insulator (SOI) wafer with 220 nm top silicon layer and 2 μm buried oxide (BOX) layer. An 1 × 2 multimode interference (MMI) is used as the splitter/combiner. The 3 dB ports of MMI are connected to 2 straight waveguides besides the micro-ring. The SEM of SSC is shown in the insert of the Fig. 2(d), which has a cantilever structure and is tilted to the edge of the silicon chip to reduce the reflection occurring between the SOA and the silicon resonator [13,14]. Besides, the SSC can enlarge the mode size of the optical field and effectively reduce the coupling loss. The radius and the free-spectral range (FSR) of the micro-ring is 5.3 μm and 19.5nm, respectively. The gain chip has facet reflectivity of 10% and 0.001%, respectively. And the gain chip also have an angled facet at one end, the optical axis of this laser becomes tilted. The combination of single MRR and the etalon forms a high-Q tunable filter which can select a single longitudinal mode from the high density of fundamental Fabry-Perot modes created by a relatively long cavity. Lasing wavelength is determined at the wavelength where the transmission peaks of MRR and etalon matched. By thermally adjusting the resonating wavelength of the MRR, the matched transmission peak is shifted, and therefore, the lasing wavelength is tuned. In order to obtain high coupling efficiency between gain chip and silicon micro-ring chip, a dual-lens coupling system is introduced. We stabilized the operating temperature of the laser by using thermo-electric cooler (TEC).

 figure: Fig. 1

Fig. 1 Schematic layout and photograph of the proposed laser.

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 figure: Fig. 2

Fig. 2 (a) Schematic layout of the silicon micro-ring chip, and (b)-(d) Photograph of the GC, MRR and SSC (the insert photograph is the SEM of the SSC).

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2.2 Coupling efficiency analysis

An efficient coupling between gain chip and silicon micro-ring chip is essential for the proposed laser. The mode field of the silicon micro-ring chip inside the SSC can be simulated by using beam propagation method (BPM). The beam divergence angle of the SSC is 14° × 19°(FWHM). While the beam divergence of the gain chip is 16°(lateral) and 30°(transverse), this mode size mismatch will lead to large cavity loss. So a couple of collimating lenses with an effective focal length of 0.5mm and 0.7mm are used for coupling the silicon micro-ring chip and gain chip to increase the coupling efficiency and the alignment tolerance. The coupling loss is estimated to be 1.3 dB between the gain chip and the silicon chip. The 1dB loss tolerances of axial, lateral and vertical alignment are 7 µm, ± 1.8 µm and ± 0.8 µm, respectively. Due to the mature lens packaging technology, the whole cavity loss including coupling loss is about 5 dB, which is measured with a tunable laser source and an optical circulator.

2.3 Simulation of the device

In this paper, a multimode rate equation is applied to analyze this proposed laser. The gain region of the device is modeled by multimode rate equation, while the micro-ring resonator and etalon are modeled by scattering matrix method. Finally, we can simulate the performance of the device by combining the two modeling methods together.

The carrier density in the laser is described by the multimode rate equation. The electron density and photon density of the mth mode are described as follow [15]:

dNdt=IeVNτBN2CN3mgmvgSm
dSmdt=ΓvggmSm+ΓvggmVvgαmSm

Where N and Sm are the mth mode electron and photon density, respectively. I, e are the injection current and the free electron charge, respectively. V is the volume of the active region. The parameters τ, B, C stand for the electron lifetime, bimolecular and Auger recombination coefficients, respectively.

The optical gain is modeled by [15,16]

g(ω0)=gN(ω0)ln(Nm/N0)/(1+εmSm)

Where N0, ε are the transparency carrier density and the gain suppression coefficient, respectively. However for a widely tunable laser, flat gain spectrum should be replaced bygN(ω0)=g0H(ω), H(ω) is a Lorentzian response function:

H(ω)=11+(ωω0)2/Δω2

Where ω0 is the peak frequency of the gain profile and Δω is the full width at half maximum (FWHM) of the gain width.

The loss of the device is given by:

αm=αa+αplpla+1lalog(1|routreff|)

Where la and lp are the length of the gain and passive part and αa and αp are the loss of the gain and passive part, respectively. rout is the reflection of the output port and reff is the effective reflection coefficient. In our laser, the effective reflection coefficient which include the effect of micro-ring and etalon is given as follow:

reff=TetalonRringTetalon

For the micro-ring resonator, the frequency domain spectrum is obtained by scattering matrix method. The reflection coefficient of the single micro-ring is given as follow [17,18]:

Rring=k2exp((αr+iβr)πr)t2exp[2(αr+iβr)πr]1

Where t is the amplitude transmission coefficient and k is amplitude coupling coefficient which can be obtained from the waveguide design [18]. Here we set them as parameters and assume there is no loss in the coupling region, according to the law of conservation of energy, t2 + k2 = 1. r is the radius of the micro-ring. βr, αr are the propagation constant and loss of the bend waveguide. Besides, the phase section is calculated as the term of exp(j(αp+βp)L), which means the electric field experiences a phase delay of βp L, where the parameter βp and L are the propagation constant and propagation distance, respectively, and αp is the loss of the phase region.

The etalon can be described as standard FP etalon filter. The transmission coefficient of the etalon is given as follow [19]:

Tetalon=t1t2exp(iδ/2)1r1r2exp(iδ)

with

δ=4πneffdcosθλ

Where t1, t2, r1, r2 are the transmission coefficient and reflection coefficient of the etalon, respectively. neff, d, θ are the effective refractive index, efficient width of the etalon and the angle of incidence, respectively.

In each time step, the active part status will be updated according to Eqs. (1)-(5) by iterating from one rate equation to the other. Meanwhile, the reflection and transmission spectrum of the MRR and etalon will be calculated using Eqs. (7)-(9). And we can get the effective reflectivity coefficient reff by Eq. (6).

Figure 3 shows the principles of wavelength selection. There are three factors affect the wavelength selection: the fixed etalon, the MRR and the cavity. The red line in Fig. 3 is the MRR’s spectrum, which acts as a tunable optical filter and the period is the free spectra range FSR = λ2/nl. n is the effective index of the micro-ring and l is the round trip distance in the MRR. The green line shows a transmission spectrum of the etalon that has periodic maxima in the wavelength region, which can be tuned and fixed to ITU-T wavelengths with 100 GHz grid. By thermally adjusting the resonating wavelength of the MRR, the transmission peak is shifted to match the etalon’s transmission peak from one ITU-T wavelength to another. The total tuning range is the FSR of the MRR, which is about 20 nm. After aligning the spectrum of the MRR with etalon, we tune the unstable longitudinal modes by adjusting the phase to realize mode-hop-free operation. The change of the phase can be monitored though the change of lasing power. Lasing occurs at the etalon’s spectrum, MRR’s spectrum and the cavity mode of the device are lined up with each other in the gain spectrum region.

 figure: Fig. 3

Fig. 3 Combination effect of etalon transmission and MRR to realize a tunable laser.

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The simulated optical spectrums are shown in Fig. 4. The extreme spectrum of the tuning range is shown in Fig. 4(a), while the Fig. 4(b) shows a typical spectrum at the center of the range. As discussed before, the FSR is inversely proportional to the radius of the MRR. So in order to get a large tuning range, we will decrease the radius of the micro-ring in further study, which is limited to the manufacturing process.

 figure: Fig. 4

Fig. 4 Simulated optical spectrum, (a) the extreme spectrum of the tuning range, and (b) the typical spectrum at the center of the range.

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The wavelength locking should consider both wavelength accuracy and a reliable operation, because mode-hop is a common phenomenon in the external cavity laser. As depicted in Fig. 5(a), the wavelength should be selected with the SMSR great than 42 dB by tuning the phase. We can also adjust the phase to distinguish the two end wavelength peaks of one FSR, which is shown in Fig. 5(b). The tuning characteristic with different ITU-T channels is shown in Fig. 5(c). The wavelength tuning accuracy is less than 10 pm as shown in Fig. 5(d).

 figure: Fig. 5

Fig. 5 (a) Simulated wavelength tuning behavior during the phase tuning, (b) Distinguish the two end wavelength peaks of one FSR by phase tuning, (c) Tuning characteristics with different ITU-T channels, and (d) Wavelength tuning accuracy.

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3. Device characterization

The output power versus the current characteristics of the proposed laser is shown in Fig. 6(a). The threshold current of this tunable laser is about 65 mA with slope efficiency of 0.145 mW/mA. The output power is 29 mW when the injection current is set to 260 mA. The linewidth of this tunable laser is measured by delayed self heterodyne method with a delay fiber of 10 km length and a LiNbO3 intensity modulator to shift the carrier frequency by 200 MHz [20]. The beat signal is captured by RF spectrum analyzer and the 3-dB spectral width is less than 150 kHz as shown in Fig. 6(b). Considering the delayed self heterodyne method gives an overestimation of the linewidth due to the low-frequency flicker noise [21,22]. The linewidth should be narrower than the test value. The wavelength of this laser can be tuned by means of thermal-optic effect of thin-film metal heaters on the top of the ring. Figure 6(c) shows a tuning wavelength range of more than 17 nm with a channel spacing of 100 GHz. A high side mode suppression ratio (SMSR) over 60 dB can be observed.

 figure: Fig. 6

Fig. 6 (a) L-I curve of the proposed laser, (b) Self heterodyne linewidth spectrum, and (c) Tunning spectrum of the proposed laser.

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4. Systematic experiment

To further evaluate the performance of the laser, we implement a coherent optical transmission experiment based on CO-OFDM system which is similar to ref [23]. The transmitted signal is generated off-line by MATLAB program with a data sequence of 231-1 pseudo-random binary sequence(PRBS). An arbitrary waveform generator (AWG) is used to generate OFDM 4-QAM or 16-QAM baseband signal generator at 12 GS/s. The proposed laser and the Agilent 81940A laser are both used as the local oscillator and the coherent light source, respectively. The measured BER curves as a function of received OSNR are illustrated in Fig. 7. The insets are the 30 Gb/s and 60 Gb/s back-to-back constellation diagrams of the polarization-multiplexed OFDM 4-QAM and 16-QAM signals. Commercial narrow linewidth laser source (Agilent 81940A) is used for comparison. The implementation of the presented laser can bring similar performance compared to the commercial laser. The use of the proposed laser enables an error-free transmission. The Agilent 81940A, as a stable scientific instrument, only shows 0.2 dB higher performance at BER of 1E-3 for 16-QAM transmission.

 figure: Fig. 7

Fig. 7 OSNR versus BER of the proposed laser compared with commercial tunable light source. (The insets are constellation diagrams of the OFDM 4-QAM and 16-QAM.)

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5. Conclusions

In summary, we have demonstrated a compact silicon narrow linewidth tunable laser based on a single micro-ring resonator. A hybrid model is used in the device design optimization. The device can be placed in a 28-pin butterfly module compatible with standard ITU-T grid. The output power of this device can reach 29 mW, with a linewidth less than 150 kHz. The tuning range is 17 nm in C-band with 60 dB side-mode-suppression-ratio (SMSR). The performance of this tunable laser has also been evaluated in CO-OFDM systems. This device shows a comparable performance with the commercial narrow linewidth laser as the source in coherent transmission systems. Our future work will focus on the optimization of the MRR design to achieve broader wavelength tuning range and lower coupling loss.

References and links

1. M. Seimetz, High-Order Modulation for Optical Fiber Transmission (Springer, 2009), Ch. 7.

2. Y. Sasahata, K. Matsumoto, T. Nagira, H. Sakuma, and K. Kishimoto, M. Suzuki1, D. Suzuki1, Y. Horiguchi, M. Takabayashi, K. Mochizuki, M. Gotoda, H. Aruga, and E. Ishimura1, “Tunable 16 DFB laser array with unequally spaced passive waveguides for backside wavelength monitor,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th3A.2.

3. A. Sivananthan, H. Park, M. Lu, J. S. Parker, E. Bloch, L. A. Johansson, M. J. Rodwell, and L. A. Coldren, “Monolithic linewidth narrowing of a tunable SG-DBR Laser,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OTh3I.3. [CrossRef]  

4. Y. Qiu, X. Xiao, M. Luo, C. Li, Q. Yang, and S. Yu, “Tunable, narrow linewidth silicon microring laser source for coherent optical communications,” in Conference on Lasers and Electro-Optics, OSA Technical Digest Series (Optical Society of America, 2015), paper JTh2A.57.

5. S. Yang, Y. Zhang, D. W. Grund, G. A. Ejzak, Y. Liu, A. Novack, D. Prather, A. E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “A single adiabatic microring-based laser in 220 nm silicon-on-insulator,” Opt. Express 22(1), 1172–1180 (2014). [CrossRef]   [PubMed]  

6. A. J. Zilkie, P. Seddighian, B. J. Bijlani, W. Qian, D. C. Lee, S. Fathololoumi, J. Fong, R. Shafiiha, D. Feng, B. J. Luff, X. Zheng, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Power-efficient III-V/Silicon external cavity DBR lasers,” Opt. Express 20(21), 23456–23462 (2012). [CrossRef]   [PubMed]  

7. J. H. Lee, I. Shubin, J. Yao, J. Bickford, Y. Luo, S. Lin, S. S. Djordjevic, H. D. Thacker, J. E. Cunningham, K. Raj, X. Zheng, and A. V. Krishnamoorthy, “High power and widely tunable Si hybrid external-cavity laser for power efficient Si photonics WDM links,” Opt. Express 22(7), 7678–7685 (2014). [CrossRef]   [PubMed]  

8. T. Matsumoto, A. Suzuki, M. Takahashi, S. Watanabe, S. Ishii, K. Suzuki, T. Kaneko, H. Yamazaki, and N. Sakuma, “Narrow spectral linewidth fullband tunable laser based on waveguide ring resonators with low power consumption,” in Optical Fiber Communication Conference (Optical Society of America, 2010), paper OThQ5. [CrossRef]  

9. T. Chu, N. Fujioka, and M. Ishizaka, “Compact, lower-power-consumption wavelength tunable laser fabricated with silicon photonic-wire waveguide micro-ring resonators,” Opt. Express 17(16), 14063–14068 (2009). [CrossRef]   [PubMed]  

10. N. Kobayashi, K. Sato, M. Namiwaka, K. Yamamoto, S. Watanabe, T. Kita, H. Yamada, and H. Yamazaki, “Silicon photonic hybrid ring-filter external cavity wavelength tunable lasers,” J. Lightwave Technol. 33(6), 1241–1246 (2015). [CrossRef]  

11. A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31. [CrossRef]  

12. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006). [CrossRef]   [PubMed]  

13. Q. Fang, J. Song, X. Luo, M. Yu, G. Lo, and Y. Liu, “Mode-size converter with high coupling efficiency and broad bandwidth,” Opt. Express 19(22), 21588–21594 (2011). [CrossRef]   [PubMed]  

14. L. Jia, J. Song, T.-Y. Liow, X. Luo, X. Tu, Q. Fang, S.-C. Koh, M. Yu, and G. Lo, “Mode size converter between high-index-contrast waveguide and cleaved single mode fiber using SiON as intermediate material,” Opt. Express 22(19), 23652–23660 (2014). [CrossRef]   [PubMed]  

15. L. A. Coldren, Diode Lasers and Photonic Integrated Circuits (John Wiley and Sons, 2012), Ch. 5.

16. B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999). [CrossRef]  

17. K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006). [CrossRef]  

18. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]  

19. M. Born, Principles of Optics (Pergamon, 1965), Ch. 7.

20. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980). [CrossRef]  

21. N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010). [CrossRef]  

22. S. Camatel and V. Ferrero, “Narrow linewidth CW laser phase noise characterization methods for coherent transmission system application,” J. Lightwave Technol. 26(17), 3048–3055 (2008). [CrossRef]  

23. D. Zhang, J. Zhao, Q. Yang, W. Liu, Y. Fu, C. Li, M. Luo, S. Hu, Q. Hu, and L. Wang, “Compact MEMS external cavity tunable laser with ultra-narrow linewidth for coherent detection,” Opt. Express 20(18), 19670–19682 (2012). [CrossRef]   [PubMed]  

References

  • View by:

  1. M. Seimetz, High-Order Modulation for Optical Fiber Transmission (Springer, 2009), Ch. 7.
  2. Y. Sasahata, K. Matsumoto, T. Nagira, H. Sakuma, and K. Kishimoto, M. Suzuki1, D. Suzuki1, Y. Horiguchi, M. Takabayashi, K. Mochizuki, M. Gotoda, H. Aruga, and E. Ishimura1, “Tunable 16 DFB laser array with unequally spaced passive waveguides for backside wavelength monitor,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th3A.2.
  3. A. Sivananthan, H. Park, M. Lu, J. S. Parker, E. Bloch, L. A. Johansson, M. J. Rodwell, and L. A. Coldren, “Monolithic linewidth narrowing of a tunable SG-DBR Laser,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OTh3I.3.
    [Crossref]
  4. Y. Qiu, X. Xiao, M. Luo, C. Li, Q. Yang, and S. Yu, “Tunable, narrow linewidth silicon microring laser source for coherent optical communications,” in Conference on Lasers and Electro-Optics, OSA Technical Digest Series (Optical Society of America, 2015), paper JTh2A.57.
  5. S. Yang, Y. Zhang, D. W. Grund, G. A. Ejzak, Y. Liu, A. Novack, D. Prather, A. E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “A single adiabatic microring-based laser in 220 nm silicon-on-insulator,” Opt. Express 22(1), 1172–1180 (2014).
    [Crossref] [PubMed]
  6. A. J. Zilkie, P. Seddighian, B. J. Bijlani, W. Qian, D. C. Lee, S. Fathololoumi, J. Fong, R. Shafiiha, D. Feng, B. J. Luff, X. Zheng, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Power-efficient III-V/Silicon external cavity DBR lasers,” Opt. Express 20(21), 23456–23462 (2012).
    [Crossref] [PubMed]
  7. J. H. Lee, I. Shubin, J. Yao, J. Bickford, Y. Luo, S. Lin, S. S. Djordjevic, H. D. Thacker, J. E. Cunningham, K. Raj, X. Zheng, and A. V. Krishnamoorthy, “High power and widely tunable Si hybrid external-cavity laser for power efficient Si photonics WDM links,” Opt. Express 22(7), 7678–7685 (2014).
    [Crossref] [PubMed]
  8. T. Matsumoto, A. Suzuki, M. Takahashi, S. Watanabe, S. Ishii, K. Suzuki, T. Kaneko, H. Yamazaki, and N. Sakuma, “Narrow spectral linewidth fullband tunable laser based on waveguide ring resonators with low power consumption,” in Optical Fiber Communication Conference (Optical Society of America, 2010), paper OThQ5.
    [Crossref]
  9. T. Chu, N. Fujioka, and M. Ishizaka, “Compact, lower-power-consumption wavelength tunable laser fabricated with silicon photonic-wire waveguide micro-ring resonators,” Opt. Express 17(16), 14063–14068 (2009).
    [Crossref] [PubMed]
  10. N. Kobayashi, K. Sato, M. Namiwaka, K. Yamamoto, S. Watanabe, T. Kita, H. Yamada, and H. Yamazaki, “Silicon photonic hybrid ring-filter external cavity wavelength tunable lasers,” J. Lightwave Technol. 33(6), 1241–1246 (2015).
    [Crossref]
  11. A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
    [Crossref]
  12. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
    [Crossref] [PubMed]
  13. Q. Fang, J. Song, X. Luo, M. Yu, G. Lo, and Y. Liu, “Mode-size converter with high coupling efficiency and broad bandwidth,” Opt. Express 19(22), 21588–21594 (2011).
    [Crossref] [PubMed]
  14. L. Jia, J. Song, T.-Y. Liow, X. Luo, X. Tu, Q. Fang, S.-C. Koh, M. Yu, and G. Lo, “Mode size converter between high-index-contrast waveguide and cleaved single mode fiber using SiON as intermediate material,” Opt. Express 22(19), 23652–23660 (2014).
    [Crossref] [PubMed]
  15. L. A. Coldren, Diode Lasers and Photonic Integrated Circuits (John Wiley and Sons, 2012), Ch. 5.
  16. B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
    [Crossref]
  17. K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
    [Crossref]
  18. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
    [Crossref]
  19. M. Born, Principles of Optics (Pergamon, 1965), Ch. 7.
  20. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
    [Crossref]
  21. N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
    [Crossref]
  22. S. Camatel and V. Ferrero, “Narrow linewidth CW laser phase noise characterization methods for coherent transmission system application,” J. Lightwave Technol. 26(17), 3048–3055 (2008).
    [Crossref]
  23. D. Zhang, J. Zhao, Q. Yang, W. Liu, Y. Fu, C. Li, M. Luo, S. Hu, Q. Hu, and L. Wang, “Compact MEMS external cavity tunable laser with ultra-narrow linewidth for coherent detection,” Opt. Express 20(18), 19670–19682 (2012).
    [Crossref] [PubMed]

2015 (1)

2014 (3)

2012 (2)

2011 (1)

2010 (1)

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

2009 (1)

2008 (1)

2006 (2)

K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
[Crossref]

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

1999 (1)

B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
[Crossref]

1980 (1)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Accard, A.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Asghari, M.

Baehr-Jones, T.

Bickford, J.

Bijlani, B. J.

Bordel, D.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Bowers, J. E.

Camatel, S.

Chen, W.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Chu, T.

Chung, Y.

B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
[Crossref]

Cohen, O.

Cunningham, J. E.

Djordjevic, S. S.

Duan, G.-H.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Ejzak, G. A.

Fang, A. W.

Fang, Q.

Fathololoumi, S.

Fedeli, J.-M.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Feng, D.

Ferrero, V.

Fong, J.

Fu, Y.

Fujioka, N.

Grund, D. W.

Hammer, M.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
[Crossref]

Han, W.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Hiremath, K. R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
[Crossref]

Hochberg, M.

Hu, Q.

Hu, S.

Ishizaka, M.

Jany, C.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Jia, L.

Jones, R.

Ke, J. H.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Kikuchi, K.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Kim, B.-S.

B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
[Crossref]

Kim, S.-H.

B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
[Crossref]

Kita, T.

Kobayashi, N.

Koh, S.-C.

Krishnamoorthy, A. V.

Lamponi, M.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Lee, D. C.

Lee, J. H.

Lelarge, F.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Li, C.

Liepvre, A. L.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Lim, A. E.-J.

Lin, S.

Liow, T.-Y.

Liu, W.

Liu, Y.

Lo, G.

Lo, G.-Q.

Luff, B. J.

Luo, M.

Luo, X.

Luo, Y.

Make, D.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Man, J. W.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Messaoudene, S.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Nakayama, A.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Namiwaka, M.

Novack, A.

Okoshi, T.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Paniccia, M. J.

Park, H.

Poingt, F.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

Prather, D.

Qian, W.

Raj, K.

Sato, K.

Seddighian, P.

Shafiiha, R.

Shubin, I.

Song, J.

Stoffer, R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
[Crossref]

Thacker, H. D.

Tu, X.

Wang, L.

Wang, X.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Watanabe, S.

Xie, L.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Yamada, H.

Yamamoto, K.

Yamazaki, H.

Yang, Q.

Yang, S.

Yao, J.

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

Yu, M.

Yuan, H. Q.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Zhang, D.

Zhang, H. G.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Zhang, Y.

Zhao, J.

Zheng, X.

Zhu, N. H.

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

Zilkie, A. J.

Electron. Lett. (2)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

IEEE J. Quantum Electron. (2)

N. H. Zhu, J. W. Man, H. G. Zhang, J. H. Ke, W. Han, W. Chen, Y. Liu, X. Wang, H. Q. Yuan, and L. Xie, “Lineshape analysis of the beat signal between optical carrier and delayed sidebands,” IEEE J. Quantum Electron. 46(3), 347–353 (2010).
[Crossref]

B.-S. Kim, Y. Chung, and S.-H. Kim, “Dynamics analysis of mode-locked sampled-grating distributed bragg reflector laser diodes,” IEEE J. Quantum Electron. 35(11), 1623–1629 (1999).
[Crossref]

J. Lightwave Technol. (2)

Opt. Commun. (1)

K. R. Hiremath, R. Stoffer, and M. Hammer, “Molding of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257(5), 277–297 (2006).
[Crossref]

Opt. Express (8)

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006).
[Crossref] [PubMed]

Q. Fang, J. Song, X. Luo, M. Yu, G. Lo, and Y. Liu, “Mode-size converter with high coupling efficiency and broad bandwidth,” Opt. Express 19(22), 21588–21594 (2011).
[Crossref] [PubMed]

L. Jia, J. Song, T.-Y. Liow, X. Luo, X. Tu, Q. Fang, S.-C. Koh, M. Yu, and G. Lo, “Mode size converter between high-index-contrast waveguide and cleaved single mode fiber using SiON as intermediate material,” Opt. Express 22(19), 23652–23660 (2014).
[Crossref] [PubMed]

S. Yang, Y. Zhang, D. W. Grund, G. A. Ejzak, Y. Liu, A. Novack, D. Prather, A. E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “A single adiabatic microring-based laser in 220 nm silicon-on-insulator,” Opt. Express 22(1), 1172–1180 (2014).
[Crossref] [PubMed]

A. J. Zilkie, P. Seddighian, B. J. Bijlani, W. Qian, D. C. Lee, S. Fathololoumi, J. Fong, R. Shafiiha, D. Feng, B. J. Luff, X. Zheng, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Power-efficient III-V/Silicon external cavity DBR lasers,” Opt. Express 20(21), 23456–23462 (2012).
[Crossref] [PubMed]

J. H. Lee, I. Shubin, J. Yao, J. Bickford, Y. Luo, S. Lin, S. S. Djordjevic, H. D. Thacker, J. E. Cunningham, K. Raj, X. Zheng, and A. V. Krishnamoorthy, “High power and widely tunable Si hybrid external-cavity laser for power efficient Si photonics WDM links,” Opt. Express 22(7), 7678–7685 (2014).
[Crossref] [PubMed]

D. Zhang, J. Zhao, Q. Yang, W. Liu, Y. Fu, C. Li, M. Luo, S. Hu, Q. Hu, and L. Wang, “Compact MEMS external cavity tunable laser with ultra-narrow linewidth for coherent detection,” Opt. Express 20(18), 19670–19682 (2012).
[Crossref] [PubMed]

T. Chu, N. Fujioka, and M. Ishizaka, “Compact, lower-power-consumption wavelength tunable laser fabricated with silicon photonic-wire waveguide micro-ring resonators,” Opt. Express 17(16), 14063–14068 (2009).
[Crossref] [PubMed]

Other (8)

M. Born, Principles of Optics (Pergamon, 1965), Ch. 7.

T. Matsumoto, A. Suzuki, M. Takahashi, S. Watanabe, S. Ishii, K. Suzuki, T. Kaneko, H. Yamazaki, and N. Sakuma, “Narrow spectral linewidth fullband tunable laser based on waveguide ring resonators with low power consumption,” in Optical Fiber Communication Conference (Optical Society of America, 2010), paper OThQ5.
[Crossref]

M. Seimetz, High-Order Modulation for Optical Fiber Transmission (Springer, 2009), Ch. 7.

Y. Sasahata, K. Matsumoto, T. Nagira, H. Sakuma, and K. Kishimoto, M. Suzuki1, D. Suzuki1, Y. Horiguchi, M. Takabayashi, K. Mochizuki, M. Gotoda, H. Aruga, and E. Ishimura1, “Tunable 16 DFB laser array with unequally spaced passive waveguides for backside wavelength monitor,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th3A.2.

A. Sivananthan, H. Park, M. Lu, J. S. Parker, E. Bloch, L. A. Johansson, M. J. Rodwell, and L. A. Coldren, “Monolithic linewidth narrowing of a tunable SG-DBR Laser,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OTh3I.3.
[Crossref]

Y. Qiu, X. Xiao, M. Luo, C. Li, Q. Yang, and S. Yu, “Tunable, narrow linewidth silicon microring laser source for coherent optical communications,” in Conference on Lasers and Electro-Optics, OSA Technical Digest Series (Optical Society of America, 2015), paper JTh2A.57.

L. A. Coldren, Diode Lasers and Photonic Integrated Circuits (John Wiley and Sons, 2012), Ch. 5.

A. L. Liepvre, C. Jany, A. Accard, M. Lamponi, F. Poingt, D. Make, F. Lelarge, J.-M. Fedeli, S. Messaoudene, D. Bordel, and G.-H. Duan, “Widely wavelength tunable hybrid III–V/silicon laser with 45 nm tuning range fabricated using a wafer bonding technique,” in 9th IEEE International Conference on Group IV Photonics (IEEE, 2012), pp. 29–31.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic layout and photograph of the proposed laser.
Fig. 2
Fig. 2 (a) Schematic layout of the silicon micro-ring chip, and (b)-(d) Photograph of the GC, MRR and SSC (the insert photograph is the SEM of the SSC).
Fig. 3
Fig. 3 Combination effect of etalon transmission and MRR to realize a tunable laser.
Fig. 4
Fig. 4 Simulated optical spectrum, (a) the extreme spectrum of the tuning range, and (b) the typical spectrum at the center of the range.
Fig. 5
Fig. 5 (a) Simulated wavelength tuning behavior during the phase tuning, (b) Distinguish the two end wavelength peaks of one FSR by phase tuning, (c) Tuning characteristics with different ITU-T channels, and (d) Wavelength tuning accuracy.
Fig. 6
Fig. 6 (a) L-I curve of the proposed laser, (b) Self heterodyne linewidth spectrum, and (c) Tunning spectrum of the proposed laser.
Fig. 7
Fig. 7 OSNR versus BER of the proposed laser compared with commercial tunable light source. (The insets are constellation diagrams of the OFDM 4-QAM and 16-QAM.)

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

dN dt = I eV N τ B N 2 C N 3 m g m v g S m
d S m dt =Γ v g g m S m + Γ v g g m V v g α m S m
g( ω 0 )= g N ( ω 0 )ln( N m / N 0 )/( 1+ε m S m )
H( ω )= 1 1+ (ω ω 0 ) 2 /Δ ω 2
α m = α a + α p l p l a + 1 l a log( 1 | r out r eff | )
r eff = T etalon R ring T etalon
R ring = k 2 exp(( α r +i β r )πr) t 2 exp[2( α r +i β r )πr]1
T etalon = t 1 t 2 exp(iδ/2) 1 r 1 r 2 exp(iδ)
δ= 4π n eff dcosθ λ

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