Semiclassical theory of noise in multielement semiconductor lasers

We present a derivation of the noise spectra of multielement semiconductor lasers. We model the noise by a set of Langevin sources which drive a system of small-signal field equations. The Langevin sources are normalized to transition rates within the laser and general formulas for relative intensity, frequency fluctuation, and field spectra are produced. We evaluate the formulas for several specific cases of interest, including those of a passive-active resonator and active-active coupled cavity resonator. In each case, the linewidth is governed by effective α-parameter(s) which generally differ from the material parameter. In the active-active cavity, the linewidth consists of two parts, one which is similar to the Schawlow-Townes linewidth, and a second which is proportional to the FM modulation index.     

Phase and amplitude fluctuations in a mode-locked laser

Comparisons are made of the phase and amplitude fluctuations of a laser when it operates single mode and when it is mode locked with the same total average power. Despite the much lower signal-to- (spontaneous emission) noise ratio of the mode-locked laser, the linewidth of each of the locked modes is the same as that of the single mode. The fluctuation of the total intensity of the mode-locked laser, and the linewidth enhancement factor due to intensity fluctuation as recently analyzed by Henry, are the same in both cases.     

Spectral properties of a semiconductor laser coupled to a single mode fiber resonator

The effect of optical feedback from a single mode fiber resonator on the instantaneous frequency fluctuation spectrum and on the field fluctuation spectrum of an AlGaAs semiconductor laser is studied theoretically and experimentally. Single mode fiber resonators of different lengths and finesses are used for the experiments. The minimum linewidth conditions are given and it is shown that the linewidth reduction ratio increases with the fiber resonator selectivity. The limit for multistability operation depends on the resonator finesse. Theory and experiments are found to be in good agreement. Finally, the effect of resonator loaded semiconductor laser phase noise on BER performance in optical DPSK heterodyne systems is theoretically analyzed.     

被动锁模Cr~(4+)∶YAG激光器的噪声分析

锁模固体激光器的激光输出脉冲的频率和强度随激光器的噪声出现非常大的波动,限制了激光器系统在许多方面的应用。理论上分析了被动锁模Cr4+∶YAG激光器的噪声形式和产生原因,从被动锁模Cr4+∶YAG激光器的脉冲传播方程出发,得到了带噪声的非线性薛定谔方程,依据这个方程的类孤子解,推导出被动锁模Cr4+∶YAG激光器腔内在噪声影响下脉冲的振幅和相位波动的线性扰动方程,对这个方程作了逼近解,得出振幅和相位波动的表达式和相应的噪声谱,可对实际的被动锁模Cr4+∶YAG激光器输出脉冲的噪声进行控制。     

The effect of spatially dependent temperature and carrier fluctuations on noise in semiconductor lasers

The spatially dependent equations of motion for a single-mode semiconductor laser including Langevin source terms are derived and solved. The relative intensity, frequency, and field fluctuation spectra are derived and calculated. The results include low-frequency excess noise, frequency noise enhancement due to two forms of amplitude-phase coupling, and power-independent contributions to the linewidth.     

Injection-Locked Mode-Locked Laser With Long-Term Stabilization and High Power-per-Combline

A novel method is presented to provide long-term stabilization of an injection-locked harmonically mode-locked laser to a narrow linewidth continuous-wave (CW) source for the first time. The method, based on the Pound-Drever-Hall method of laser frequency stabilization, exploits the phase shift across a laser cavity mode resonance to detect the offset between the CW frequency and the combline frequency desired for injection-locking. Long-term suppression of the radio-frequency supermode noise spurs is shown as well as 36-dB suppression of optical supermodes. Significant reductions in phase and amplitude noise are also shown due to injection. Postamplification noise results are presented, showing an increase in optical power per combline with no degradation in pulse-to-pulse energy fluctuation or timing jitter.     

Quantitative Study of Optical Frequency Noise to Intensity Noise Conversion in Line-by-Line Pulse Shaping

We report the first quantitative study of intensity noise induced in line-by-line pulse shaping in response to time-varying changes in the comb frequency offset. Controllable comb linewidth broadening is synthesized through frequency dithering of a continuous-wave laser that is fed to a phase modulator. An electrical spectrum analyzer is used to examine the current power spectra of shaped time-domain intensity waveforms subject to comb frequency noise. A theoretical model predicting a 20 dB/decade scaling relation between the dither-induced noise and the frequency dither amplitude is presented. A numerical simulation method capable of predicting the precise form of the RF power spectrum in the presence of optical frequency dithering is explained. Two line-by-line shaping cases are considered in detail. Experimental data are in excellent agreement with the simulated results down to frequency dithers of a few tenths of a percent of the comb spacing. Tolerances to laser frequency fluctuations are given for several simple pulse shaping examples. The effect of pulse shaper parameters is also discussed.      

Numerical simulation of intensity and phase noise from extracted parameters for CW DFB lasers

A self-consistent numerical approach is demonstrated to analyze intensity and phase noise from experimentally extracted parameters for a continuous-wave distributed feedback (DFB) laser. The approach takes into account the intrinsic fluctuations of the photon number, carrier number, and phase. Values for the parameters appearing in the rate equations are extracted from the measured relative intensity noise spectra and linewidth of the laser. The simulation of the frequency spectra of intensity and phase noise of the DFB laser are performed by fast Fourier transform and exhibit good agreement with experimental results. The model presented here can readily be extended for the purpose of system simulations.     

The effect of side modes with linewidth and intensity fluctuationsin semiconductor lasers

The influence of side modes on the linewidth and intensity fluctuations of the main mode is studied using Monte Carlo techniques. Explicit analytic relations are developed for the side-mode induced linewidth, using impulse responses of the laser obtained from rate equations. It is found that both the absolute level of the main-mode too side-mode power ratio and the passive loss in the side mode influence the main-mode linewidth. A typical 3-mW laser has negligible linewidth enhancement from the side mode if the modal power ratio is about 300/1 (or greater), corresponding to a side-mode loss of 0.2-0.5 dB (or more) for one round trip within the laser resonator. Lower side-mode losses yield an abrupt catastrophic increase in main-mode linewidth     

Nonlinear dynamics of a laser diode with optical feedback systemssubject to modulation

The nonlinear dynamic behavior of a direct frequency-modulated diode laser with strong optical feedback is examined and compared to a laser diode subject to electro-optically modulated, strong optical feedback. Direct modulation is achieved by sinusoidal modulation of the diode laser injection current. Electro-optic modulation is achieved by applying a sinusoidal voltage to an intracavity phase modulating element. The output state (characterized by the output power versus time, the intensity noise spectrum and the optical frequency spectrum) for both types of modulation is dependent on the ratio of the modulation frequency to the external cavity resonant frequency, and the modulation power. A number of distinct states are observed: conventional amplitude modulation (with FM spectra); multimode, low-noise amplitude modulation; multimode, high-noise amplitude modulation; periodic limit-cycle operation; quasi-periodicity; chaos; low-frequency fluctuations; and mode-locking. There are significant differences between the direct and electro-optic frequency-modulation cases. The onset of the dynamic instability is characterized as a noisy period-one oscillation for direct modulation and a low-frequency fluctuation for intracavity electro-optic modulation. Phase portraits produced experimentally with the use of a digital phosphor oscilloscope are shown to agree well with those constructed from output power versus time data. This represents an experimental method for examining the dynamics phase portraits in real-time     

交叉关联噪声驱动的单模激光模型稳态分析：修正的郎之万方程情形

本文对具有加性和乘性白噪声关联的单模激光模型的统计性质进行了研究。从一个具有交叉关联噪声与单模激光与立方模随机等价郎之万方程，计算了稳态激光光强度的平均值、协方差和偏斜率。并与早期结果比较，阐明了考虑郎之万方程修正项对描述激光系统统计性质的重要性。     

Nonlinear distortions and noise in optical communication systems due to fiber connectors

The transmission characteristics of optical fiber connectors are analyzed in detail in order to calculate nonlinear distortions and noise due to instable speckle patterns. The fluctuation amplitude of the transmission loss and its sensitivity, with respect to a wavelength shift of the laser source, are determined for coherent and partially coherent laser sources. Nonlinear distortions due to fiber connectors yield a second-order harmonic distortion of typically -36 dB for a single-longitudinal mode injection laser, and typically -65 dB for a multimode laser. The noise performance has also been estimated for both low frequency and high frequency fluctuations.     

大气信道下窄线宽激光器对相干通信系统性能影响研究

在空间下行相干激光通信系统中,激光器的线宽大小会影响通信系统的性能,而窄线宽激光器能有效降低激光器线宽引起的激光器相位噪声,目前已经成为相干激光通信系统的首选。光信号在大气信道传输时,大气湍流会引起光信号强度和相位的波动,从而进一步影响系统的通信性能。针对上述问题,基于四相移键控(Quadri Phase Shift Keying, QPSK)通信系统的工作原理,进一步考虑窄线宽激光器线宽和大气湍流引起的光信号强度和相位的波动,给出了空间下行QPSK通信系统误码率模型。并基于该模型,数值仿真分析了窄线宽激光器对空间下行相干激光通信系统性能的影响。结果表明：大气湍流不仅严重影响系统性能,也会弱化激光器线宽对系统性能的影响。而对于大气湍流影响而言,其引起的相位波动要大于光强波动的影响。此外,随着通信速率增大,激光器线宽对于系统性能影响也会随之降低。文中对于空间下行相干激光通信系统的优化设计和调试具有一定的实际参考意义。     

Noise of mode-locked lasers

A theory of noise in mode-locked lasers is developed that applies to additive pulse mode-locked and Kerr lens mode-locked systems. Equations of motion are derived for pulse energy, carrier linewidth, frequency pulling, and timing jitter. The effect of gain fluctuations, mirror vibrations, and index fluctuations are determined. Measurements that can determine all four fluctuation spectra are described. Experimental data in the literature are compared with theory     

Realization of ultrahigh coherence in semiconductor lasers bynegative electrical feedback

To be useful, an ultrahigh-coherence semiconductor laser source requires high frequency stability, narrow linewidth, the capability of frequency tracking to a master laser, and stable frequency tuning. Negative electrical feedback is proposed to meet these four requirements simultaneously. Although the degree of frequency fluctuation that can be reduced by negative electrical feedback is limited by the noise contained in the feedback signal, theoretical calculations show that the fluctuations can be lower than the quantum noise limit. Experimental results obtained recently by the author are reviewed     

Numerical modeling of intensity and phase noise in semiconductorlasers

A self-consistent numerical approach is demonstrated to analyze intensity and phase noise in semiconductor lasers. The approach takes into account the intrinsic fluctuations of the photon number, carrier number, and phase. A new systematic technique is proposed to generate the Langevin noise sources that derive the laser rate equations keeping their cross-correlations satisfied. The simulation is applied to AlGaAs lasers operating in a single mode. The time-varying profiles of the fluctuating photon and carrier numbers and the instantaneous shift of the oscillating frequency are presented. Statistical analysis of the intensity and phase fluctuations is given. The frequency spectra of intensity and phase noise are calculated with help of the fast Fourier transform. The importance of taking into account the carrier number noise source and its cross-correlation with the noise source on the phase is examined by comparing our results with those by conventional methods     

Noise spectra and noise correlation functions of the single-modelaser with a low-Q cavity

The noise characteristics of a single-mode laser with a low-Q cavity are investigated theoretically. After the electric field is adiabatically eliminated from the Maxwell-Bloch equations, coupled Langevin equations with both additive and multiplicative white noises are examined. The equations are solved using Rice's method in a framework of quasilinear Fourier analysis. Noise spectral densities are calculated analytically to study the dependence of the relative intensity noise (RIN) on the pumping. Through an investigation of the auto- and cross-correlations of the light intensity and population noises, their variances are obtained in order to compare their properties to the good-cavity case. The stationary intensity cumulants, the photon counting coefficient, and the photon counting probability are explicitly derived and compared with the results of a Fokker-Planck analysis previously carried out for both the good- and bad-cavity cases     

Measurement of laser quantum frequency fluctuations using aPound-Drever stabilization system

We describe a method for measuring the frequency fluctuation spectrum of a laser oscillator, especially the weak noise contributions in the wings of the spectrum, and apply this method to confirm the existence of large excess quantum frequency fluctuations in a laser oscillator using an unstable optical resonator. Our measurement apparatus uses the Pound-Drever technique, which employs an RF phase modulator and a Fabry-Perot cavity to produce a sensitive high-speed frequency discrimination signal. We show that this signal can also be used to measure the quantum noise contributions to the frequency spectrum of a laser oscillator. Experimental measurements on a miniature diode-pumped Nd:YAG laser using a stable optical cavity closely match the predictions of the usual Schawlow-Townes theory, while the frequency fluctuations in a nearly identical laser employing an unstable optical resonator are approximately 1300 times larger. These much larger fluctuations arise in part from the larger output coupling and cavity bandwidth of the unstable cavity, but they also appear to confirm a predicted excess spontaneous emission factor (Petermann excess noise factor) of ≈180 times arising from the nonorthogonal transverse mode properties of the unstable cavity     

输入信号后单模激光系统线性化近似适用范围分析

运用线性化近似方法计算泵噪声和实虚部间关联的量子噪声驱动的单模激光系统输入信号后的稳态平均光强关联函数和稳态平均光强相对涨落，对所用线性化近似方法的适用范围进行了详细分析，具体讨论了量子噪声实虚部间关联系数、量子噪声强度、泵噪声强度、输入信号振幅和频率、净增益等对稳态平均光强相对涨落的影响，发现在量子噪声实虚部间弱关联、小噪声、远离阈值、信号强度不大和频率较高的条件下，线性化近似有较高精度。     

Modulation performance of a semiconductor laser coupled to anexternal high-Q resonator

The dynamic response of a semiconductor laser coupled to an external resonator is studied using the single-mode rate equations modified to account for the dispersive feedback. Both the frequency and the damping rate of relaxation oscillations are affected by the feedback. The frequency chirp that invariably accompanies amplitude modulation is significantly reduced. The feedback also reduces the phase noise and the linewidth. To investigate the usefulness of external-resonator lasers in high-speed optical communication systems, the rate equation have been solved numerically to obtain the emitted chirped pulse; the pulse is propagated through the fiber, detected, and filtered at the receiver. The simulated-eye diagrams show that such lasers can be operated at high bit rates with negligible dispersion penalty owing to their reduced frequency chip