Christopher R. Jones and J. Nathan Kutz, "Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method," J. Opt. Soc. Am. B 27, 1184-1194 (2010)

The stability of local structures in optical systems is of great importance. We demonstrate that using the Floquet–Fourier–Hill (FFH) method provides a substantial improvement in both speed and accuracy over finite-difference methods that are commonly used. Furthermore, we show how to incorporate the effect of nonlocal saturable gain in the linearization and stability predictions. Several examples of problems are worked with both the FFH and finite-difference methods and compared in the context of mode-locked laser models.

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Comparison Between the FFH and FD Calculations for the Phase Sensitive Amplifier Linear Operator^{
a, b
}

Max $\left|{\lambda}_{0}\right|$

FFH Method

FD Method

Min m

${\lambda}_{\text{max}}$

$\Delta t\left(\text{s}\right)$

Min m

${\lambda}_{\text{max}}$

$\Delta t\left(\text{s}\right)$

${10}^{-2}$

57

0.238 36

0.1

254

0.238 391

0.3

${10}^{-4}$

119

0.238 387 10

0.5

782

0.238 387 20

4.7

${10}^{-6}$

157

0.238 387 161 6

0.8

2476

0.238 387 16

9.3

${10}^{-8}$

191

0.238 387 162 2

1.1

—

—

—

${10}^{-10}$

225

0.238 387 162 2

1.4

—

—

—

${10}^{-12}$

283

0.238 387 162 2

2.8

—

—

—

The parameters chosen are ${g}_{0}=1.0081$, $\beta =1$, and $\omega =1$, corresponding to $\eta =1.55$. Here the error tolerance refers to the magnitude of the zero eigenvalue $\left|{\lambda}_{0}\right|$, m is the minimum matrix dimension needed to achieve the desired tolerance, ${\lambda}_{\text{max}}$ is the eigenvalue with the largest real part, and $\Delta t$ is the central processing unit time needed to build the matrix and calculate the eigenvalues in MATLAB using the eig command.
A dash in the table indicates that the matrix could not be built given the memory constraints.

Table 2

Comparison of Eigenvalue with the Maximum Real Part Using FFH and FD Methods for Varying Grid Sizes^{
a, b
}

N

Saturable Gain

Fixed Gain

FFH

FD

FFH

FD

31

$1.3\times {10}^{-1}$

$3.5\times {10}^{-1}$

0.30

0.41

63

$8.9\times {10}^{-4}$

$1.5\times {10}^{-1}$

0.1980

0.15

127

$1.2\times {10}^{-9}$

$4.0\times {10}^{-2}$

0.197 793 123 2

0.194

255

$3.4\times {10}^{-14}$

$3.9\times {10}^{-3}$

0.197 793 123 845 3

0.1976

511

$2.7\times {10}^{-14}$

$2.6\times {10}^{-4}$

0.197 793 123 845 4

0.1978

N refers to the number of points in the discretization of the real and imaginary parts separately, so the total matrix size is $2N$.
In each case the computational domain was taken to be from $-15$ to 15. In this case the saturating gain is responsible for stabilizing the solution.

Table 3

Comparison of the Two Zero Eigenvalues Using the FFH and FD Methods for Varying Grid Sizes^{
a, b
}

N

FFH

FD

${\lambda}_{0,1}$

${\lambda}_{0,2}$

${\lambda}_{0,1}$

${\lambda}_{0,2}$

31

$1.3\times {10}^{-1}$

$-2.7\times {10}^{-1}$

$-0.04+0.78i$

$-0.04-0.78i$

63

$8.9\times {10}^{-4}$

$\left(-3.3+3.1i\right)\times {10}^{-1}$

$-0.04+0.11i$

$-0.04-0.11i$

127

$1.2\times {10}^{-9}$

$-1.6\times {10}^{-7}$

$-8.8\times {10}^{-3}$

$3.4\times {10}^{-2}$

255

$\left(-3.2+1.1i\right)\times {10}^{-14}$

$\left(2.7+0.1i\right)\times {10}^{-13}$

$-5.1\times {10}^{-4}$

$3.9\times {10}^{-3}$

511

$\left(2.66-0.08i\right)\times {10}^{-14}$

$\left(-3.2-2.2i\right)\times {10}^{-14}$

$-3.2\times {10}^{-5}$

$2.6\times {10}^{-4}$

N refers to the number of points in the discretization of the real and imaginary parts separately, so the total matrix size is $2N$.
In each case the computational domain was taken to be from $-15$ to 15.

Table 4

Comparison of the Two Zero Eigenvalues using the FFH and FD Methods for Varying Grid Sizes of the Waveguide Array for Parameters Corresponding to $\eta =1.83$^{
a, b, c
}

N

FFH

FD

${\lambda}_{0,1}$

${\lambda}_{0,2}$

${\lambda}_{0,1}$

${\lambda}_{0,2}$

15

$-6.5$

7.2

$-0.8+9.9i$

$-0.8-9.9i$

31

5.9

$-6.0$

$-0.8+9.9i$

$-0.8-9.9i$

63

0.8

$-3.2$

$-0.9+6.4i$

$-0.9-6.4i$

127

$3.5\times {10}^{-4}$

$-0.01$

$-0.48$

0.96

255

$\left(3.8+0.1i\right)\times {10}^{-13}$

$-3.9\times {10}^{-9}$

$-0.028$

0.100

511

$\left(-1.9+0.1i\right)\times {10}^{-12}$

$\left(3.0-1.4i\right)\times {10}^{-12}$

$-0.0018$

0.0067

In each case N refers to the number of fourier modes for the real and imaginary parts of each waveguide, so the total matrix size is $6N$.
As can be seen from the table, the FFH method calculates the two known zero eigenvalues to $\approx {10}^{-9}$ accuracy using 255 Fourier modes, and ${10}^{-13}$ with 511 modes, at which point the accuracy is limited by numerical effects.
Larger matrices could not be studied using MATLAB owing to memory constraints. Bearing in mind these results, we chose $N=255$ for all WGA calculations.

Tables (4)

Table 1

Comparison Between the FFH and FD Calculations for the Phase Sensitive Amplifier Linear Operator^{
a, b
}

Max $\left|{\lambda}_{0}\right|$

FFH Method

FD Method

Min m

${\lambda}_{\text{max}}$

$\Delta t\left(\text{s}\right)$

Min m

${\lambda}_{\text{max}}$

$\Delta t\left(\text{s}\right)$

${10}^{-2}$

57

0.238 36

0.1

254

0.238 391

0.3

${10}^{-4}$

119

0.238 387 10

0.5

782

0.238 387 20

4.7

${10}^{-6}$

157

0.238 387 161 6

0.8

2476

0.238 387 16

9.3

${10}^{-8}$

191

0.238 387 162 2

1.1

—

—

—

${10}^{-10}$

225

0.238 387 162 2

1.4

—

—

—

${10}^{-12}$

283

0.238 387 162 2

2.8

—

—

—

The parameters chosen are ${g}_{0}=1.0081$, $\beta =1$, and $\omega =1$, corresponding to $\eta =1.55$. Here the error tolerance refers to the magnitude of the zero eigenvalue $\left|{\lambda}_{0}\right|$, m is the minimum matrix dimension needed to achieve the desired tolerance, ${\lambda}_{\text{max}}$ is the eigenvalue with the largest real part, and $\Delta t$ is the central processing unit time needed to build the matrix and calculate the eigenvalues in MATLAB using the eig command.
A dash in the table indicates that the matrix could not be built given the memory constraints.

Table 2

Comparison of Eigenvalue with the Maximum Real Part Using FFH and FD Methods for Varying Grid Sizes^{
a, b
}

N

Saturable Gain

Fixed Gain

FFH

FD

FFH

FD

31

$1.3\times {10}^{-1}$

$3.5\times {10}^{-1}$

0.30

0.41

63

$8.9\times {10}^{-4}$

$1.5\times {10}^{-1}$

0.1980

0.15

127

$1.2\times {10}^{-9}$

$4.0\times {10}^{-2}$

0.197 793 123 2

0.194

255

$3.4\times {10}^{-14}$

$3.9\times {10}^{-3}$

0.197 793 123 845 3

0.1976

511

$2.7\times {10}^{-14}$

$2.6\times {10}^{-4}$

0.197 793 123 845 4

0.1978

N refers to the number of points in the discretization of the real and imaginary parts separately, so the total matrix size is $2N$.
In each case the computational domain was taken to be from $-15$ to 15. In this case the saturating gain is responsible for stabilizing the solution.

Table 3

Comparison of the Two Zero Eigenvalues Using the FFH and FD Methods for Varying Grid Sizes^{
a, b
}

N

FFH

FD

${\lambda}_{0,1}$

${\lambda}_{0,2}$

${\lambda}_{0,1}$

${\lambda}_{0,2}$

31

$1.3\times {10}^{-1}$

$-2.7\times {10}^{-1}$

$-0.04+0.78i$

$-0.04-0.78i$

63

$8.9\times {10}^{-4}$

$\left(-3.3+3.1i\right)\times {10}^{-1}$

$-0.04+0.11i$

$-0.04-0.11i$

127

$1.2\times {10}^{-9}$

$-1.6\times {10}^{-7}$

$-8.8\times {10}^{-3}$

$3.4\times {10}^{-2}$

255

$\left(-3.2+1.1i\right)\times {10}^{-14}$

$\left(2.7+0.1i\right)\times {10}^{-13}$

$-5.1\times {10}^{-4}$

$3.9\times {10}^{-3}$

511

$\left(2.66-0.08i\right)\times {10}^{-14}$

$\left(-3.2-2.2i\right)\times {10}^{-14}$

$-3.2\times {10}^{-5}$

$2.6\times {10}^{-4}$

N refers to the number of points in the discretization of the real and imaginary parts separately, so the total matrix size is $2N$.
In each case the computational domain was taken to be from $-15$ to 15.

Table 4

Comparison of the Two Zero Eigenvalues using the FFH and FD Methods for Varying Grid Sizes of the Waveguide Array for Parameters Corresponding to $\eta =1.83$^{
a, b, c
}

N

FFH

FD

${\lambda}_{0,1}$

${\lambda}_{0,2}$

${\lambda}_{0,1}$

${\lambda}_{0,2}$

15

$-6.5$

7.2

$-0.8+9.9i$

$-0.8-9.9i$

31

5.9

$-6.0$

$-0.8+9.9i$

$-0.8-9.9i$

63

0.8

$-3.2$

$-0.9+6.4i$

$-0.9-6.4i$

127

$3.5\times {10}^{-4}$

$-0.01$

$-0.48$

0.96

255

$\left(3.8+0.1i\right)\times {10}^{-13}$

$-3.9\times {10}^{-9}$

$-0.028$

0.100

511

$\left(-1.9+0.1i\right)\times {10}^{-12}$

$\left(3.0-1.4i\right)\times {10}^{-12}$

$-0.0018$

0.0067

In each case N refers to the number of fourier modes for the real and imaginary parts of each waveguide, so the total matrix size is $6N$.
As can be seen from the table, the FFH method calculates the two known zero eigenvalues to $\approx {10}^{-9}$ accuracy using 255 Fourier modes, and ${10}^{-13}$ with 511 modes, at which point the accuracy is limited by numerical effects.
Larger matrices could not be studied using MATLAB owing to memory constraints. Bearing in mind these results, we chose $N=255$ for all WGA calculations.