The Covariance Matrix Method

1
The Covariance Matrix Method and Its Relation
to Soliton Perturbation Theory presented
by Curtis R. Menyuk with B. S. Marks and J.
Zweck
2
The Covariance Matrix Method and Its Relation
to Soliton Perturbation Theory presented
by Curtis R. Menyuk and thanks to R.
Holzlöhner, V. S. Grigoryan, W. L. Kath
3
Uniting three conceptual themes

• Soliton Perturbation Theory? Applications to
  optical systems
• Noise? Applications to communication systems,
  lasers
• Computer Algorithms? Quantitative modeling of
  complex systems

4
Perturbation Theory

• Our basic transmission equation nonlinear
  Schrödinger equation
• Additive
• We write and obtain
• a deterministic (nonlinear) piece
• a noise-driven (linear) piece

noise perturbation
5
Perturbation Theory

• Additive
• Only small time shifts can be accounted for.
  Example
• if

6
Perturbation Theory

• Soliton
• Theory is more complex
• Theory can account for large time and phase
  shifts shape distortions should be small

7
Perturbation Theory

• Soliton
• Theory is more complex
• Theory can account for large time and phase
  shifts shape distortions should be small
• Observation Complex systems often have
  qualitatively similar behavior with different
  shapes!

8
Perturbation Theory

• Soliton
• Theory is more complex
• Theory can account for large time and phase
  shifts shape distortions should be small
• Observation Complex systems often have
  qualitatively similar behavior with different
  shapes!
• Query How do we calculate this behavior?

9
Noise

• Basic Assumption
• Noise sources are additive, white, Gaussian
  AWGN assumptionThis assumption is very good in
  most optical systems

10
Noise

• Doobs Theorem
• where
• R general linear integro-differential operator
  in time
• multivariate Gaussian source (not
  necessarily white)
• ? u( z,t ) is multivariate-Gaussian-distributed

11
Noise

• Doobs Theorem
• u( z,t ) is multivariate-Gaussian-distributed
• What does this mean?
• Discretize
  in time
• Let
• The entire distribution is determined by m and K.
  
• We only have to solve for K(z) when m 0

12
Noise

• Doobs Theorem
• u( z,t ) is multivariate-Gaussian-distributed
• What does this mean?
• The entire distribution is determined by m and K.
  
• We only have to solve for K(z) when m 0

The covariance matrix methods linearizes the
nonlinear Schrödinger equation and its
variants using soliton-like degrees-of-freedom
13
Computer Algorithm

• Solve the linearized, homogeneous propagation
  equationwhere a is the vector of Fourier
  modes
• Calculate the transformation matrix, defined
  byand the covariance matrix for the Fourier
  modesG gain, h I ASE noise contribution
  (diagonal)

14
Computer Algorithm

• Separate phase and timing jitter using two-step
  Gram-Schmidt orthogonalizationThis
  produces the revised covariance matrix K(r)

15
Computer Algorithm

• Separate phase and timing jitter using two-step
  Gram-Schmidt orthogonalization

Is this really needed?

• Yes!! Without this process resultsare
  inaccurate even in quasilinearsystems.

16
Validation

• Comparison
• Covariance matrix method (deterministic)
• Faster ? Approximate
• Multicanonical Monte Carlo (statistical)
• Slower ? Accurate to within statistical
  error
• Additional difficulty
• System complexity
• transmitter receiver error-correction
• must be analyzed together

17
Validation
Submarine single-channel 10 Gb/s CRZ system, 6120
km
916 ps/nm
916 ps/nm
34 map periods
post-compensation
pre-compensation
16.5 ps/nm-km
A
?2.5 ps/nm-km
20 km
25 km
45 km
45 km
45 km
Nonlinear scale length 1960 km System length
3 nonlinear scale lengths
18
Validation
Probability density
Voltage (normalized)
Covariance matrix method and multicanonical Monte
Carlo agree perfectly over 15 orders magnitude!
R. Holzlöhner and C. R. Menyuk, Opt. Lett. 28,
1894 (2003)
19
Future Perspectives
Powerful methods have been developed that allow
accurate calculation of noise effects in complex
systems! BUT Communications systems may not be
the best application of these ideas

• Modern optical communication systems are nearly
  linear
• Multiple pulse interactions dominate the
  nonlinearity
• The advent of FEC means that raw BERs of 103 or
  104 arecommon

Lasers may be a good application!