1. 1 Moshe Nazarathy Copyright Lecture VI Introduction to Fiber Optic Communication Optical amplifiers (ch. 16 – part 3 “Notes”) …in which we study Optical Amplification Moshe Nazarathy All Rights Reserved Ver1

2. 2 Moshe Nazarathy Copyright Optical Direct Detection

3. 3 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission

4. 4 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission (II)

5. 5 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission (III)

6. 6 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission (IV)

7. 7 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission (V)

8. 8 Moshe Nazarathy Copyright The quantum limit for ideal OOK optical transmission (VI) =10 Ten average photons per bit Twenty photons in the “one” bit

9. 9 Moshe Nazarathy Copyright Photon counting detection of optical binary transmission with two non-zero waveforms (I) “On-pulse” “Off-pulse” Self-study

10. 10 Moshe Nazarathy Copyright Photon counting detection of optical binary transmission with two non-zero waveforms (II) Log-likelihood ratio Self-study

11. 11 Moshe Nazarathy Copyright Photon counting detection of optical binary transmission with two non-zero waveforms (III) Self-study

12. 12 Moshe Nazarathy Copyright …where Poissonian Photon Statistics and Gaussian Johnson noise meet…

13. 13 Moshe Nazarathy Copyright I&D Detection in shot noise approximated as gaussian I&D filter Shot Noise Approximate the Poisson as Gaussian approx.

14. 14 Moshe Nazarathy Copyright Optical Amplification

15. 15 Moshe Nazarathy Copyright Physical principles of optical amplification Optical Amplifier Types: –Erbium Doped Fiber Amplifiers (EDFA) –Semiconductor Amplifiers –Raman Fiber Amplifiers Principle: A collection of quantum objects (atoms, electron- hole pairs, etc.) are “pumped” by an external means (optical, electrical) into excited energy states. Stimulated Emission: Incoming photons induce additional photons of the same direction, frequency and polarization. The excited entities go down to the ground state Amplified Spontaneous Emission (ASE): the excited objects randomly relax to the ground state while emitting photons. Same effects as used in lasers (laser = amplifier + feedback)

16. 16 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05 Optical Power in two-sided bandwidth B in both quadratures and one polarization = + OA Optical Power Gain Effective Noise Field Redefine E-field units such that P=|E|^2

17. 17 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05 2-sided Optical All statements here are per one polarization - The two polarizations fluctuations are i.i.d. Total power in both orthogonal polarizations is double

18. 18 Moshe Nazarathy Copyright Noise statistics of the optical amplifier + OA Spontaneous noise factor SIGNAL AWGN Two-sided PSD of the field CE (both quadratures) [Watt/Hz] When E is in units of sqrt[Watt] then PPSD coincides with the PSD of the field: PSD=Var in 1 Hz; Var = mean sq of ZM r.v. When E is in units of [ (Volt/Meter) 2 /Hz ] the PSD and PPSD are merely proportional Photonic Power Spectral Density (PPSD): The ASE optical field noise process is approximated as white (and gaussian): high gain amps Effective Noise Field

19. 19 Moshe Nazarathy Copyright Noise statistics of the optical amplifier Two-sided PSD of the real optical field and of each quadrature component [Watt/Hz + ] + OA SIGNAL AWGN Two-sided PSD of the optical field rms complex envelope: Photonic PSD One-sided PSD of the real optical field as well as of each quadrature component One-sided PSD rms complex envelope of the optical field: summary See next slide to justify the I&Q stmts

20. 20 Moshe Nazarathy Copyright PSD Spectra of narrowband white noise and its I&Q and complex representation PSD Quadrature representation From T1 ModCom

21. 21 Moshe Nazarathy Copyright Optical Amplification spontaneous noise factor derivation

22. 22 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

23. 23 Moshe Nazarathy Copyright Optical Engineering characteristics of the optical amplifier (I) we view the optical amplifier... as an amplifier, striving to derive some of its optical engineering characteristics, such as its noise figure and its RIN, as well as its CNR contribution to the output in a link which includes multiple optical amplifiers. The spontaneous spectral optical power (mean optical power in Watts per one-sided Hz) at the output of the OA in one polarization is given by: amplified emission 2-sided 2

24. 24 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05 Self-study

25. 25 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05 Self-study

26. 26 Moshe Nazarathy Copyright Optical Amplification signal-spontaneous and spontaneous- spontaneous noise contributions derivation

27. 27 Moshe Nazarathy Copyright Canonical Optical Amplification Problem: White Gaussian ASE, Flat o- and e- filters, CW signal Ideal responsivity: Set in the model of last slide: CW Signal: ASE PSD: PSD of terms: PSD of term: electr. PSD [Olsson, `89] Without proof

28. 28 Moshe Nazarathy Copyright Canonical Optical Amplification Problem: Signal-spontaneous noise dominates over shot-noise Ideal responsivity: PSD of terms: electr. PSD [Olsson, `89] Its flat in-band level: The Sig-spont. PSD is stronger than the output shot-noise PSD that’d have been generated by the same CW input signal going through an ideal OA with the same gain, then photo-detected with the same responsivity COMPARE SIG-SPONT. vs. SHOT NOISES

29. 29 Moshe Nazarathy Copyright Conventional Derivation of s-sp and sp-sp noise one-sided is

30. 30 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise -

31. 31 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise

32. 32 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise jointly

33. 33 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise

34. 34 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise again… super alias names for the signal beating with the noise

35. 35 Moshe Nazarathy Copyright Signal-spontaneous beat noise Effective IM noise source… Multiply by the responsivity to yield the s-sp induced photocurrent noise

36. 36 Moshe Nazarathy Copyright Derivation of s-sp and sp-sp noise Convolve this with itself

37. 37 Moshe Nazarathy Copyright Now we must evaluate the autoconvolution: Signal-spontaneous and spontaneous-spontaneous beat noises (VIII) The one-sided spontaneous-spontaneous noise spectral density is then | alias names for the noise beating with itself Area under this graph (multiplied with itself)

38. 38 Moshe Nazarathy Copyright Signal-spontaneous and spontaneous-spontaneous beat noises (IX) | its peak value

39. 39 Moshe Nazarathy Copyright OA Signal-spontaneous and spontaneous-spontaneous beat noises (X) Consider an optical transmission situation involving an optical amplifier.

40. 40 Moshe Nazarathy Copyright Optical Amplifier CNR and Noise Figure THIS SECTION: SELF-STUDY (its content is partially treated in TA class)

41. 41 Moshe Nazarathy Copyright Optical Amplifier CNR (I) The OA internal noise figure of the amplifier is defined as the ratio of internal input and output CNRs The input CNR (carrier-to-noise ratio) spectral density (in 1 Hz two-sided bandwidth) is referred to the ‘purest’ light, coherent light afflicted only by its inherent shot noise: OA Consider an optical transmission situation involving an optical amplifier. without the effect of coupling

42. 42 Moshe Nazarathy Copyright Here we have specifically labelled the CW power with the superscript “in”, and have expressed a generic optical shot noise as the shot noise associated with the photon stream carrying total energy “charge” To formally justify the optical shot noise expression, consider the photon arrivals’ process Optical Amplifier CNR (II) P

43. 43 Moshe Nazarathy Copyright Optical Amplifier CNR (III) ----

44. 44 Moshe Nazarathy Copyright Review the derivation why spectral density (=arrivals rate) for filtered Poisson process Optical Amplifier CNR (IV) ----

45. 45 Moshe Nazarathy Copyright Optical Amplifier CNR (V)

46. 46 Moshe Nazarathy Copyright Again the beat noise will only show up on the photo-detector, however we choose to represent it as if it is already existent in the optical domain. So, in a sense these are photocurrent mean square quantities, all referred back to the input optical domain by division by the responsivity. The output CNR then consists of the signal spectral density over the output shot noise plus output s-sp contributions: taking the ratio between the input CNR, and the output CNR…. Optical Amplifier CNR (VI)

47. 47 Moshe Nazarathy Copyright Optical Amplifier noise figure (I) ----

48. 48 Moshe Nazarathy Copyright This indicates that When ideally i.e. the ideal large-gain amp noise figure is 3 dB Optical Amplifier noise figure (II)

49. 49 Moshe Nazarathy Copyright PAM Detection with optically pre-amplified receiver

50. 50 Moshe Nazarathy Copyright Beat-noise-limited OA Let the ASE (amplified spontaneous emission) be the dominant noise component All other noise sources in optical detection may be neglected. Realistic assumption for a receiver with an optical pre-amplifier with sufficient optical gain and reasonable noise figure the thermal noise photocurrent is negligible relative to the signal-spontaneous noise component in the photocurrent. The ASE noise is usually modelled as AWGN

51. 51 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05 Conventional OSNR definition is not very precise… (though related)

52. 52 Moshe Nazarathy Copyright OSNR – relation to Symbol-SNR “Logical 1” symbol energy Symbol period (inverse of baudrate) 2 factor due to two pol. in one pol. See next page for OOK ASE PWR in a reference 0.1 nm band =0.1 nm at 1.5um

53. 53 Moshe Nazarathy Copyright

54. 54 Moshe Nazarathy Copyright Noise statistics of the optical amplifier Two-sided PSD of the real optical field and of each quadrature component [Watt/Hz + ] + OA SIGNAL AWGN Two-sided PSD of the optical field rms complex envelope: Photonic PSD One-sided PSD of the real optical field as well as of each quadrature component One-sided PSD rms complex envelope of the optical field: summary See next slide to justify the I&Q stmts

55. 55 Moshe Nazarathy Copyright two pol. OSNR – relation to Symbol-SNR (II) Shown Later This one-sided PSD formula for the the real field in one polarization (=2-sided PSD for the CE field in one polarization) but beware – the OSNR denom has noise for two polarizations one pol. related to NF # of photons per (1) symbol at OA input Redefine E-field units such that:

56. 56 Moshe Nazarathy Copyright The pulse energy is then Modeling optically-amplified direct detection Let the optical field (noiseless signal) response at the OA output due to a lone transmitted symbol be expressed in terms of the unit symbol response from the transmitted symbol to the OA output + OA FIBER CHANNEL OPT TX Unit symbol response: Two-sided PSD of each ASE quadrature

57. 57 Moshe Nazarathy Copyright Modeling Optically-Amplified Direct Detection + OA FIBER CHANNEL OPT TX OF photons per pulse The symbol SNR at the OA output equals the ratio of the mean photon count at the OA input and the spontaneous noise factor: OADD Proof: THEOREM: is the two-sided PSD of each ASE quadrature watts more precisely G-1: corr. factor G/(G-1)

58. 58 Moshe Nazarathy Copyright Modeling Optically-Amplified Direct Detection + OA FIBER CHANNEL OPT TX OF Photons per pulse @OA input “energy” in the communication sense (mean square integral of the amp. waveform) OF photo-det sq. env-det =“Symbol SNR” @ OA output “Electrical” equiv. comm. system: If OF is a matched filter: For stat. analysis purposes we may set Phase uncertainty in laser source and fiber optical path length + A responsivity gain would not matter: signal & noise equally amplified No electrical filter assumed hence squaring & sampling may commute With MF, SamplerSNR=2*SymbolSNR “Sampler SNR” at MF output:

59. 59 Moshe Nazarathy Copyright Indep. PAM ASYNCH (NON-COHERENT / ENVELOPE) DETECTION PAM Coherent/Non-Coherent Bandpass channel – complex representation SYNCHONOUS (COHERENT) DETECTION ISI-Free From S3 Goto S3->… …->Return

60. 60 Moshe Nazarathy Copyright Indep. PAM ASYNCH (ENVELOPE) DETECTION Coherent/Non-Coherent Bandpass channel – complex representation ISI-Free Signal space (complex plane) OOK modulation format: On: Off: On: Off: matched OF Evaluate “sampler SNR” (Q-sq.): constant gain factors in f(t) do not matter

61. 61 Moshe Nazarathy Copyright Indep. Coherent/Non-Coherent Bandpass channel – complex representation ISI-Free Signal space (complex plane) On: Off: OOK PAM "Sampler SNR” (Q-squared factor) =twice the “symbol SNR” (also from matched filtering theory): Received effective # of photons at OA input in the on-pulse per quadrature dimension

62. 62 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation On: Off: OOK PAM Decision law: Right tail of Rician distrib. Marcum-Q function Rice pdf shown last slide

63. 63 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation On: Off: OOK PAM Decision law: (Rice with a=0) Alternatively,, is chi-squared with two DOFs, i.e. exponentially distributed

64. 64 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation On: Off: OOK PAM For high SNR it may be shown that the optimal threshold tends to half-way between on and off in the amplitude domain

65. 65 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation On: Off: OOK PAM Assume general threshold

66. 66 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation + OA FIBER CHANNEL OPT TX OF photons on average photo-det Cf. OOK with ideal photon counting: More precise threshold optimization (file PhIM.nb & next page): Literature quotes as it uses a discrete photon arrivals model for the photodetector (and evaluates OA output statistics) rather than assuming a quadratic detection model like here

67. 67 Moshe Nazarathy Copyright Coherent/Non-Coherent Bandpass channel – complex representation + OA FIBER CHANNEL OPT TX OF Photons in “on” pulse photo-det This may be alternatively derived applying the Normalized Q-factor Model (next slide):

68. 68 Moshe Nazarathy Copyright ASYN PAM PWEP- Normalized Q-factor (NQF) (i) ASYN On-Off Keying (OOK): …continued… Assume MF: Review from S4 Modern Comm

69. 69 Moshe Nazarathy Copyright Exercise: Evaluate the BER for a two-level ASK modulation format: The upper level has K photons at the OA in The lower level has photons where is the extinction ratio.

70. 70 Moshe Nazarathy Copyright Example: duobinary with optical amplification Exercise: Derive an expression for the BER of duobinary detection with optical amplification. Cf. [Bosco, et. al]

71. 71 Moshe Nazarathy Copyright OVERFLOWS

72. 72 Moshe Nazarathy Copyright Optical Engineering characteristics of the optical amplifier (II) “1Hz” always 2-sided in these slides SKIP

73. 73 Moshe Nazarathy Copyright Optical Engineering characteristics of the optical amplifier (III) So we can relate the optical spectral power to the (two-sided) spectral density of the field: “1Hz” always 2-sided in these slides + OA for large gain In a noiseless amplifier of gain if we inject a beam with Poisson statistics with two-sided spectral density thus with the same average count per second (see next slide) then the ideally amplified input noise generates at the output the same average count as the actual noisy amp, Ideally, for the input referred effective uncertainty representing the ASE output noise is one photon per sec Also think in terms of an effective noisy input field 2-s

74. 74 Moshe Nazarathy Copyright Reminder: PSD of a homogeneous Poisson impulsive process e.g. a photons arrivals process for a CW beam (each photon arrival marked by an impulse) Stationary (homogenous) Poisson process with rate (mean number of arrivals per second) We have shown (in “Random Signals”) that the spectrum of this process is white. Assuming whiteness, its particular PSD constant level is readily derived as follows: Let us filter the process through an LTI “spectral-analysis” filter with unity gain and 1 Hz two-sided bandwidth, or equivalently with unity energy The PSD level equals the variance at the output of this filter. E.g., a one-second integrator, with impulse response has unity energy, unity DC gain, 1 Hz effective noise bandwidth. Therefore its output variance, when is input into it, equals But the integrator output also describes the number of arrivals in 1 sec, hence is Poisson distributed, i.e. the mean count equals the variance. On the other hand, the mean count at the output equals the process rate We conclude: Two-sided PSD level = the mean count rate

75. 75 Moshe Nazarathy Copyright Optical Engineering characteristics of the optical amplifier (IV) SKIP

76. 76 Moshe Nazarathy Copyright Alternative derivation of the optical amplifier noise figure THIS SECTION: SELF-STUDY

77. 77 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

78. 78 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

79. 79 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

80. 80 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

81. 81 Moshe Nazarathy Copyright Optical Engineering Application: Optical Amplifiers Cascade THIS SECTION: SELF-STUDY

82. 82 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

83. 83 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

84. 84 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

85. 85 Moshe Nazarathy Copyright Mecozzi, ECOC ‘05

86. 86 Moshe Nazarathy Copyright Statistical optical mixing of stationary independent light sources A useful analysis for optical measurements Here it provides an alternative derivation of the signal x spontaneous beat noise THIS SECTION: SKIP

87. 87 Moshe Nazarathy Copyright Beating two light beams (I) Expressed in terms of the analytic signals and complex envelopes, the electric fields are:

88. 88 Moshe Nazarathy Copyright Beating two light beams (II)

89. 89 Moshe Nazarathy Copyright Beating two light beams (III)

90. 90 Moshe Nazarathy Copyright Beating two light beams (IV)

91. 91 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (I) The resulting theory will be useful in the analysis of optical measurements as well as in figuring out the noise properties of optical amplifiers.

92. 92 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (II)

93. 93 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (III) 2

94. 94 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (IV) Stationary whenever the fields are The time-averaged autocorrelation of the mixing term of the irradiance for a totally general superposition of two independent fields simplifies to:

95. 95 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (V) At this point, let us specialize to two independent stationary fields, in which case we have seen the fourth moment of the field breaking down into a product of two t- invariant autocorrelations. Application of the time averaging then has no effect in this case, yielding the following expression for the mixing term of the irradiance generated by the superposition of two stationary independent fields:

96. 96 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (VI) stationary

97. 97 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (VII) stationary

98. 98 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (VIII) repeat… deterministic cross-correlation of two waveforms

99. 99 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (IX) Opt. freq. RF freq.

100. 100 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (X) The total irradiance of the superposition of the two independent stationary fields. We identify nine terms in its autocorrelation: a a *

101. 101 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (XI)

102. 102 Moshe Nazarathy Copyright Second-order statistics of the superposition of two light fields (XII) independent 1 1 -T/2 T/2 Self-study The paper also treats the case when one or both of the beating fields is nonstationary, as is the case when one or both of the fields are subject to modulation.

103. 103 Moshe Nazarathy Copyright Signal-spontaneous beat noise derivation (I) We have worked hard in section ? to derive the heterodyne beating term between two mutually incoherent optical signals that superpose together. Applying those results to the problem at hand, we view the amplifier output as the superposition of the signal and ASE fields: The total photocurrent is given by sig-spont. noise spont-spont. noise

104. 104 Moshe Nazarathy Copyright To begin with, let us take for simplicity It is convenient to revert to the respective irradiances in the optical domain, doing away with the scaling constants These should be considered as representations of the respective photocurrents, linearly mapped back for convenience to the irradiance domain. Signal-spontaneous beat noise derivation (II)

105. 105 Moshe Nazarathy Copyright ?? Signal-spontaneous beat noise derivation (III)

106. 106 Moshe Nazarathy Copyright Signal-spontaneous beat noise derivation (IV)

107. 107 Moshe Nazarathy Copyright Apply theorem on the spectrum of the irradiance of the superposition of two independent fields: Yielding for the cross-term: Signal-spontaneous beat noise derivation (V)

108. 108 Moshe Nazarathy Copyright MID1 MID2 Signal-spontaneous beat noise derivation (VI)

109. 109 Moshe Nazarathy Copyright Signal-spontaneous beat noise derivation (VII)

110. 110 Moshe Nazarathy Copyright OA Signal-spontaneous and spontaneous-spontaneous beat noises – summary

111. 111 Moshe Nazarathy Copyright Signal-Spontaneous and Spontaneous-Spontaneous beat noise

112. 112 Moshe Nazarathy Copyright Signal-spontaneous and spontaneous-spontaneous beat noises (I) We view the amplifier output as the superposition of the signal and ASE fields: The total photocurrent is given by sig-spont. noise spont-spont. noise

113. 113 Moshe Nazarathy Copyright Signal-spontaneous and spontaneous-spontaneous beat noises (II) To begin with, let us take for simplicity It is convenient to revert to the respective irradiances in the optical domain, doing away with the scaling constants These should be considered as representations of the respective photocurrents, linearly mapped back for convenience to the squared optical field domain (or to the optical power domain). Alternatively, we might work in the optical power domain: It is as if there are optical power fluctuations of these sizes… Effective IM noise terms…

114. 114 Moshe Nazarathy Copyright Signal-spontaneous and spontaneous-spontaneous beat noises (II) Again the beat noise terms will only show up on the photo-detector. However we choose to represent it as if it is already existent in the optical domain. In a sense these are photocurrent mean square fluctuations, all referred back to the input optical domain by division by the responsivity.

115. 115 Moshe Nazarathy Copyright Optically Pre-Amplified Receiver Introduce the cross-correlation of deterministic real waveforms: independent in the general results Set PSD of term: PSD of terms: Assume stationary, Gaussian: skip

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