Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN Lecture 19- ECE 240a Laser Phase Noise 1 ECE 240a Lasers - Fall 2019 Lecture 19
Phase Noise Lecture 19- ECE 240a Phase Noise Phase Noise Assume laser is operating well above threshold. Power Spectrum RIN Affect of spontaneous emission is to “de-phase” the carrier without affecting the amplitude. Study amplitude (intensity noise) later Field given by � I ( t ) e j ( 2 πf c t + φ ( t )) U ( t ) = where for now we only assume φ ( t ) is random. Form optical autocorrelation � ( U ( t ) U ∗ ( t + τ )) � R ( τ ) = √ I o e j ( 2 πf c t + φ ( t )) √ I o e − j ( 2 πf c ( t + τ )+ φ ( t + τ )) � = � P o e j 2 πf c τ � e j ( φ ( t ) − φ ( t + τ )) � . = 2 ECE 240a Lasers - Fall 2019 Lecture 19
Ideal Laser Source Lecture 19- ECE 240a Phase Noise We first consider an idealized deterministic signal with P ( t ) = P and Phase Noise Power φ ( t ) = φ o . Spectrum RIN The autocorrelation function for the laser is � ( s ( t ) s ∗ ( t + τ )) � . R ( τ ) = √ Pe i φ √ Pe − i φ � = � = P The autocorrelation function is equal to the constant power P . The corresponding power density spectrum is S ( f ) = Pδ ( f − f c ) Some laser sources can approach this limit Many laser diode sources are far from this limit having unmodulated spectral widths that are a significant fraction of the carrier frequency 3 ECE 240a Lasers - Fall 2019 Lecture 19
Model for Phase Noise Lecture 19- ECE 240a Model for laser is laser has a constant amplitude and a random phase. Phase Noise Phase Noise Power Spectrum The justification for treating the amplitude as a constant when deriving RIN the laser power density spectrum: Phase fluctuations within an oscillator require no net energy transfer, Amplitude fluctuations require a net energy transfer Therefore, random phase fluctuations are the dominant physical mechanism affecting the power density spectrum of oscillators. The time-varying phase fluctuations produce a frequency deviation, f d , from the carrier frequency f d ( t ) = 1 dφ ( t ) 2 π dt Physically, the frequency deviation is related to the rate of spontaneous emission. Each spontaneous emission event perturbs the laser frequency in a random fashion. 4 ECE 240a Lasers - Fall 2019 Lecture 19
Frequency Noise Lecture 19- ECE 240a Given that there are many independent perturbing events, we can assert Phase Noise the central limit theorem and model the random frequency deviation Phase Noise Power process f d ( t ) as a zero-mean, gaussian noise process with an Spectrum autocorrelation function RIN R ( τ ) = � f d ( t ) f d ( t + τ ) � = Kδ ( τ ) The constant power density spectrum K of the frequency deviation is related to the rate of spontaneous emission. The impulsive form of the autocorrelation function is indicative of a process where the observation time τ is much longer than the correlation time of the random frequency deviations. The phase in the integral of this white noise process. This is called is a Wiener process given by the integral of the gaussian random process f d ( t ) � t φ ( t ) = 2 π f d ( τ ) dτ 0 5 ECE 240a Lasers - Fall 2019 Lecture 19
Plot of the Random Walk for the Phase Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN Phase Time 6 ECE 240a Lasers - Fall 2019 Lecture 19
Expression for Autocorrelation Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN Expression for autocorrelation R ( τ ) = P o exp [ j 2 πν 0 τ ] � exp [ j ( φ ( t + τ ) − φ ( t ))] � − 2 π 2 K | τ | � P o exp [ j 2 πν 0 τ ] exp � = Form is double-sided exponential. 7 ECE 240a Lasers - Fall 2019 Lecture 19
Power Spectrum Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN Power spectrum is Fourier transform of autocorrelation function B o / 2 π S ( f ) = P ( f − f c ) 2 + ( B o / 2 ) 2 where B o = πK is the optical half-power point. This width is called the intrinsic linewidth of the laser. This is the spectral width of the laser for a constant injection current. 8 ECE 240a Lasers - Fall 2019 Lecture 19
Plot of R ( τ ) and S ( f ) Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN 9 ECE 240a Lasers - Fall 2019 Lecture 19
Comments Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum As the rate K of spontaneous emission increases, the linewidth broadens RIN because there are more spontaneous emission events per unit time and each event slightly perturbs the laser frequency. A more detailed analysis reveals that the spontaneous emission rate K reciprocally scales with the mean number of photons S in the resonator so that K ∝ 1 / S. The mean photon number depends on both the power and the quality of the resonator. As S increases, the ratio of the stimulated emission to the spontaneous emission increases, the laser becomes more coherent, and the bandwidth B o decreases. Therefore, resonator structures with long photon lifetimes are desired to produce narrow linewidth lasers. 10 ECE 240a Lasers - Fall 2019 Lecture 19
Comments-2 Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN The inverse relationship of the linewidth to the power in the resonator is valid up to moderate power levels. For higher powers, the linewidth can rebroaden and the linewidth can increase because of other effects not incorporated into the model. The spectral width also increases if the amplitude and phase fluctuations are coupled within the resonator as is the case for semiconductor lasers. The intrinsic linewidth defines the coherence of the laser. 11 ECE 240a Lasers - Fall 2019 Lecture 19
Intensity Noise Lecture 19- ECE 240a Phase Noise Phase Noise Power Spectrum RIN Define normalized optical power autocorrelation function r P ( τ ) = � ∆ P ( t ) ∆ P ( t + τ ) � � P � 2 where ∆ P = P − � P � is the variation of the lightwave signal power about the mean � P � . At τ = 0 , this correlation function becomes σ 2 r P ( τ )( 0 ) = � ∆ P 2 � = � P 2 � − � P � 2 1 P = � P � 2 = � P � 2 � P � 2 SNR 12 ECE 240a Lasers - Fall 2019 Lecture 19
Relative Intensity Noise (RIN) Lecture 19- ECE 240a Phase Noise Fourier transform of R P o ( τ ) is defined to the relative intensity noise Phase Noise Power Spectrum spectrum RIN ( f ) RIN � ∞ r P ( τ )( τ ) e − j 2 πfτ dτ RIN ( f ) = 2 −∞ where the factor of two converts a two-sided power density spectrum to a one-sided power density spectrum. Power density spectrum from the RIN is given by N RIN ( f ) = R 2 � P � 2 RIN ( f ) The RIN can be directly measured using a calibrated detector with known noise characteristics and an electrical spectrum analyzer (ESA). The units of RIN are typically dB/Hz where dB is w/respect to the average power at a specific frequency measured over a 1 Hz bandwidth. 13 ECE 240a Lasers - Fall 2019 Lecture 19
Excess RIN Lecture 19- ECE 240a The minimum amount of intensity noise is generated when the laser is Phase Noise operating at the shot noise limit. Phase Noise Power Spectrum Setting N RIN ( f ) equal to the shot noise power density N shot = 2 e � i � RIN and using an ideal sensor with a quantum efficiency η = 1 , N shot R� P � = 2 hf 2 e � P � = 2 e RIN shot ( f ) = R 2 � P � 2 = � i � e where � i � is the mean sensed signal, and R = hf is the responsivity of an ideal sensor with η = 1 . This is the minimum amount of relative intensity noise that can be generated by any lightwave source. Additional power fluctuations above this minimum level are called excess relative intensity noise. The intensity noise spectrum RIN ( f ) has a � P � − 1 dependence similar to that of the phase noise because as the source power increases, there is more stimulated emission relative to spontaneous emission and thus less noise emitted from the laser. 14 ECE 240a Lasers - Fall 2019 Lecture 19
Intensity Noise Statistics Lecture 19- ECE 240a Phase Noise Distribution for intensity noise can be written as Phase Noise Power Spectrum 2 1 � − ( P − v ) 2 � f P ( P ) = √ π 1 + erf ( v ) exp P ≥ 0 RIN P = P / ( √ πP th ) is a normalized power v is an inversion parameter that varies from large negative numbers below threshold, to zero at threshold, to large values above threshold. The parameter P th is the threshold power of the mode. For v ≪ 0 , which corresponds to a laser far below the threshold for oscillation, the distribution approaches an exponential distribution f P ( P ) ≈ 2 | v | exp [ − 2 | v | P ] P ≥ 0 and the intensity noise statistics are pseudo-thermal with a mean value � P � = √ πP th / 2 | v | 15 ECE 240a Lasers - Fall 2019 Lecture 19
Plot of Distribution Lecture 19- ECE 240a Phase Noise Phase Noise 0 Power Log probability distribution Spectrum � 2 RIN � 4 � 6 (a) (b) (c) � 8 � 10 � 12 � 14 0 2 4 6 8 10 12 P/ ( √ πP th ) For v ≫ 0 , which corresponds to a laser operating well above threshold, the distribution approaches a Gaussian distribution 2 � − ( P − v ) 2 � f P ( P ) = √ π exp P ≥ 0 with a mean value � P � = √ πP th v and a variance σ 2 = √ πP th / 4 v . 16 ECE 240a Lasers - Fall 2019 Lecture 19
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