in Sources Power Factor Limit Optical ECE 185 Lasers and Modulators Lab - Spring 2018 1
in Sources Power Factor Limit We treat noise on a per mode basis Total noise is then noise/mode number of modes An estimate of the number of noise modes is the bandwidth Fundamental quantum noise energy per mode given by E = hf 2 Caused by interaction with vacuum fluctuations. Note power density spectrum (Watts/Hz is a unit of energy J/s/Hz=J) so that mean noise energy equivalent to mean noise power per unit bandwidth Fundamental noise equivalent to 1 2 photon per mode Source is fundamental fluctuations in all electromagnetic modes called vacuum fluctuations ECE 185 Lasers and Modulators Lab - Spring 2018 2
Fundamental Sources of in Sources Power Factor Limit (N(f)/kT0) Power 100 10 1 10 1 10 2 10 3 30000 10 10 3000 10 11 Wavelength in microns 300 10 12 10 13 sum 10 3 10 2 10 1 1 10 Photon Energy in units of kt 30 Frequency in hertz hf 2kT 0 hf/kt 0 exp(hf/kt 0 ) 1 3 0.3 10 14 10 15 100 ECE 185 Lasers and Modulators Lab - Spring 2018 3
Amplifier in Sources Power Factor Limit Amplifiers add additional noise to fundamental quantum noise Mean noise energy also 2 1 photon per mode Total noise referred to the input of the amplifier is then 1 photon per mode We then express gain equation in terms of photons per mode as d n s ( ) = γ int ns + 1 dz where n s is average number of signal photons per mode Term γ int is gain from stimulated emission not including loss ECE 185 Lasers and Modulators Lab - Spring 2018 4
Amplifier Loss in Sources Power Factor Limit When there is loss (absorption) signal is attenuated. is also attenuated, but as it is absorbed, it produces new noise. Net result is signal is attenuated, but not total noise. Equation for signal loss is then d n s = α n s dz ECE 185 Lasers and Modulators Lab - Spring 2018 5
Gain Equation with in Sources Power Factor Limit Add two equation together to get gain equation with noise d n s = γ n s + γint dz where γ = γ int α is the net gain. Solve small-signal gain equation with input being n in ns = G nin + nsp where n sp is the photon noise density. ECE 185 Lasers and Modulators Lab - Spring 2018 6
Power in Sources Power Factor Limit Term n sp is the photon noise density nsp = γ int (G 1) γ = n sp(g 1) Corresponding noise power density spectrum scaled by energy of photon hf N sp(f) = hfn sp(g 1) W/Hz. Total noise power σsp 2 (W) is the integral over the band σsp 2 = N sp(f)df 0 ECE 185 Lasers and Modulators Lab - Spring 2018 7
Factor in Sources Power Factor Limit Term n sp is spontaneous emission noise factor n sp = γ int γ γ = int γ int α σ en = 2 σ en 2 σ an 1 1 Equality approached if σ en 2 σ an 1 When n sp 1, amplifier generates minimum amount of noise. ECE 185 Lasers and Modulators Lab - Spring 2018 8
Limit in Sources Power Factor Limit When gain is high (G 1) and noise is low n sp 1 then noise power density is N sp(f) hfg Corresponds to the equivalent of one spontaneously emitted photon per mode at the input to the optical amplifier - quantum noise limit. Minimum amount of spontaneous emission that can be generated by the amplifier. ECE 185 Lasers and Modulators Lab - Spring 2018 9
Power in Sources Power Factor Limit Express one photon per mode in units of energy per mode or in units of power/frequency Energy per mode = Power Frequency = Energy/time = Energy 1/time Energy using a wavelength of λ = 1.5 microns (200 THz) is then E = hf = hc λ = (6.62 10 34 )(3 10 8 ) 1.5 10 6 = 1.325 10 19 J Express in terms of dbm/hz (dbm is power in db referenced to 1 mw) E = 10 log(1.325 10 19 ) + }{{} 30 ref. to 1 mw = 159 dbm/hz Note that thernal noise (kt ) at room temperature is 174 dbm/hz 15 db larger (32 ) larger than thermal noise ECE 185 Lasers and Modulators Lab - Spring 2018 10
Fundamental Sources of in Sources Power Factor Limit Power (units of kt) 1000. 100. 10. 1. 0.1 0.01 0.001 Light@ 1.5 microns (200 Thz) 0.0001 0.0003 0.003 0.03 0.3 3. 30. 300. Frequency for temperature of 300 K (GHz) ECE 185 Lasers and Modulators Lab - Spring 2018 11
(outline) in Sources Power Factor Limit Optical power is the square of the optical signal plus noise. If signals and noise expressed in complex notation, then power P is P (t) = 1 2 s(t) + n(t) 2 where s(t) = 2P s(t) and n(t) = 2P n(t)e jφ(t) are the lightwave signal and noise complex amplitudes. Expand out the expression P (t) = 1 2 s(t) + n(t) 2 = = 2 1 } s(t) 2 {{} signal power + s(t)n(t) }{{} signal-noise beat + 1 2 n(t) 2 }{{} noise-noise beat Output spectrum is then (derived in detail in notes on Web) S P (f) = P 2 δ(f) + 2 P s Sn(f) + S n(f) S n(f). where S n(f) is noise power density spectrum. ECE 185 Lasers and Modulators Lab - Spring 2018 12
Example in Sources Power Factor Limit Let S n(f) = N sprect(f/b o) Then S P (f) = P 2 δ(f) + 2 P s Nsprect(f/B o) + 2N 2 sp (1 f /B o), There is additional shot noise for both the signal and the noise. Sensed output signal after detection is S y(f) = R 2 P 2 H(0) 2 δ(f) }{{} mean signal + R P H(f) 2 }{{} shot noise +R 2 2P sn sprect(f/b o) + 2N 2 sp (1 f /Bo) H(f) 2 } {{ } sig.-spont. noise }{{} spont.-spont.-noise ECE 185 Lasers and Modulators Lab - Spring 2018 13
Plot of Terms in Sources Power Factor mean signal Limit spont. -spont. noise shot noise sig.-spont. noise f c f ECE 185 Lasers and Modulators Lab - Spring 2018 14
Total Power in Sources Power Factor Limit Total electrical power given by integral ( P e = S y(f)df = R 2 2 [ ] ) 2 P H(0) + 2eR P B N + R 2 4P sn spb N + 4N 2 sp BoB N where H(f) 2 df = 2B N is defined as the electrical noise bandwidth. dbm Hz 130 140 150 160 170 180 190 Sig Sp Total Sp Sp Thermal Total Shot 200 40 30 20 10 0 10 Amplified Signal Power dbm ECE 185 Lasers and Modulators Lab - Spring 2018 15
in Sources Power Factor Limit Concept of noise figure used to characterize noise performance of optical amplifiers Standard definition is ratio of the input Signal-to--Ratio (SNR in ) at the input to the amplifier to the output Signal-to--Ratio (SNR out ) where the signal and noise are powers which is the square of the signal. Most OSAs use a definition of optical power as the signal and then square this Assume input is shot noise limited so that photon noise power (variance) is equal to mean (Poisson statistics.) Therefore SNR in = n 2 σ 2 n = (ns/g)2 n s/g = ns/g ECE 185 Lasers and Modulators Lab - Spring 2018 16
Cont. in Sources Power Factor Limit Output SNR using power is then SNR out = ns2 σ 2 n where σ 2 n is the photon number variance after amplification. figure is the ratio F NP = SNR in = ns/g SNR out n 2 s /σn 2 = 1 + 2 ( ) n sp nsp 1 + nsp + G Gn s ECE 185 Lasers and Modulators Lab - Spring 2018 17
in Terms of Power in Sources Power Factor Limit Using the following conversions to get noise figure in terms of what the OSA measures nsp = P aset /hf = P ase/2b reshf n s = GP in /2B reshf where B res = 1/2T is the resolution bandwidth of the OSA. Making these substitutions, the noise figure is F NP = SNR in = SNR out 1 G }{{} sig. shot noise + P ase GP in }{{} ase shot noise P ase Pase 2 + + hfgb res 2hfG 2 B resp in }{{} sig.-ase noise } {{ } ase-ase noise This expression is what the OSA in lab measures. ECE 185 Lasers and Modulators Lab - Spring 2018 18