Lecture 19- ECE 240a Laser Phase Noise 1 ECE 240a Lasers - Fall - - PowerPoint PPT Presentation

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

RIN

Lecture 19- ECE 240a

Laser Phase Noise

ECE 240a Lasers - Fall 2019 Lecture 19 1

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

RIN

Phase Noise

Assume laser is operating well above threshold. Affect of spontaneous emission is to “de-phase” the carrier without affecting the amplitude. Study amplitude (intensity noise) later Field given by U(t) =

• I(t)ej(2πfct+φ(t))

where for now we only assume φ(t) is random. Form optical autocorrelation R(τ) = (U(t)U∗(t + τ)) =

• √

Ioej(2πfct+φ(t))√ Ioe−j(2πfc(t+τ)+φ(t+τ)) = Poej2πfcτej(φ(t)−φ(t+τ)).

ECE 240a Lasers - Fall 2019 Lecture 19 2

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

RIN

Ideal Laser Source

We first consider an idealized deterministic signal with P(t) = P and φ(t) = φo. The autocorrelation function for the laser is R(τ) = (s(t)s∗(t + τ)). =

• √

Peiφ√ Pe−iφ = P The autocorrelation function is equal to the constant power P. The corresponding power density spectrum is S(f) = Pδ(f − fc) 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

ECE 240a Lasers - Fall 2019 Lecture 19 3

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Model for Phase Noise

Model for laser is laser has a constant amplitude and a random phase. The justification for treating the amplitude as a constant when deriving 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, fd, from the carrier frequency fd(t) = 1 2π dφ(t) 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.

ECE 240a Lasers - Fall 2019 Lecture 19 4

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Frequency Noise

Given that there are many independent perturbing events, we can assert the central limit theorem and model the random frequency deviation process fd(t) as a zero-mean, gaussian noise process with an autocorrelation function R(τ) = fd(t)fd(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 fd(t) φ(t) = 2π

t

fd(τ)dτ

ECE 240a Lasers - Fall 2019 Lecture 19 5

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Plot of the Random Walk for the Phase

Time Phase

ECE 240a Lasers - Fall 2019 Lecture 19 6

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Expression for Autocorrelation

Expression for autocorrelation R(τ) = Po exp [j2πν0τ] exp [j (φ(t + τ) − φ(t))] = Po exp [j2πν0τ] exp −2π2K|τ| Form is double-sided exponential.

ECE 240a Lasers - Fall 2019 Lecture 19 7

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Power Spectrum

Power spectrum is Fourier transform of autocorrelation function S(f) = P Bo/2π (f − fc)2 + (Bo/2)2 where Bo = 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.

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Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Plot of R(τ) and S(f)

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Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Comments

As the rate K of spontaneous emission increases, the linewidth broadens 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 Bo decreases. Therefore, resonator structures with long photon lifetimes are desired to produce narrow linewidth lasers.

ECE 240a Lasers - Fall 2019 Lecture 19 10

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Comments-2

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.

ECE 240a Lasers - Fall 2019 Lecture 19 11

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Intensity Noise

Define normalized optical power autocorrelation function rP (τ) = ∆P(t)∆P(t + τ) P2 where ∆P = P − P is the variation of the lightwave signal power about the mean P. At τ = 0, this correlation function becomes rP (τ)(0) = ∆P 2 P2 = P 2 − P2 P2 = σ2

P

P2 = 1 SNR

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Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Relative Intensity Noise (RIN)

Fourier transform of RPo(τ) is defined to the relative intensity noise spectrum RIN(f) RIN(f) = 2

∞

−∞

rP (τ)(τ)e−j2πfτdτ where the factor of two converts a two-sided power density spectrum to a

• ne-sided power density spectrum.

Power density spectrum from the RIN is given by NRIN(f) = R2P2RIN(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.

ECE 240a Lasers - Fall 2019 Lecture 19 13

Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

RIN

Excess RIN

The minimum amount of intensity noise is generated when the laser is

• perating at the shot noise limit.

Setting NRIN(f) equal to the shot noise power density Nshot = 2ei and using an ideal sensor with a quantum efficiency η = 1, RINshot(f) = Nshot R2P2 = 2e RP = 2hf P = 2e i where i is the mean sensed signal, and R =

e 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.

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Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Intensity Noise Statistics

Distribution for intensity noise can be written as fP(P) = 2 √π 1 1 + erf(v) exp

• − (P − v)2

P ≥ 0 P = P /(√πPth) 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 Pth is the threshold power of the mode. For v ≪ 0, which corresponds to a laser far below the threshold for

• scillation, the distribution approaches an exponential distribution

fP(P) ≈ 2 |v| exp [−2 |v| P] P ≥ 0 and the intensity noise statistics are pseudo-thermal with a mean value P = √πPth/2 |v|

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Lecture 19- ECE 240a Phase Noise

Phase Noise Power Spectrum

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Plot of Distribution

Log probability distribution (a) (b) (c) P/(√πPth)

2 4 6 8 10 12 14 12 10 8 6 4 2

For v ≫ 0, which corresponds to a laser operating well above threshold, the distribution approaches a Gaussian distribution fP(P) = 2 √π exp

• − (P − v)2

P ≥ 0 with a mean value P = √πPthv and a variance σ2 = √πPth/4v.

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