Inferring the time-dependent complex Ginzburg-Landau equation from modulus data

We present a formalism for inferring the equation of evolution of a complex wave field that is known to obey an otherwise unspecified $(2+1)$-dimensional time-dependent complex Ginzburg-Landau equation, given field moduli over various closely spaced planes. The phase of the complex wave field is retrieved via a noninterferometric method, and all terms in the equation of evolution are determined using only the magnitude of the complex wave field. The formalism is tested using simulated data for a generalized nonlinear system with a single-component complex wave field. The method can be generalized to multicomponent complex fields.

Figure 1

The evolution of the complex field $\Psi$ as a function of the evolution parameter $z$. We use the intensity $I$ at various slices $z=z_0,z=z_2$, and $z=z_4$ to infer the evolution equation of the system.

Figure 2

The rms error of the phase gradient $\sigma \left(\mid {\mathbf{\nabla }}_{\perp }\Phi \mid \right)$ as a function of the iteration number $k$. (a) $A_{\Phi }=1$ (with $⟨\mid \mathbf{\nabla }{\Phi }_E\mid ⟩=0.982$), showing that the error decreases rapidly and hence the numerical scheme converges. (b) $A_{\Phi }=2$ (with $⟨\mid \mathbf{\nabla }{\Phi }_E\mid ⟩=1.97$). Here the error oscillates, illustrating that the numerical scheme does not converge. After about 390 iterations the error in (b) diverges.

Figure 3

(a) Retrieved phase at $z_1(A_{\Phi }=0.1)$ and (b) relative error in the phase. The phase is retrieved from the simulated intensity data via the multigrid iterative phase retrieval scheme using the intensity at $z_0$ and $z_2$. The relative error in the retrieved phase indicates that phase retrieval is less accurate at the boundary where $I\to 0$; however, the relative error is small in the interior (where $I$ deviates away from zero). This illustrates that it is possible to recover the phase accurately, given modulus information.

Figure 4

Relative error in the retrieved phase with various errors in the diffusion parameter $\eta$. (a) A negative error in $\eta$ gives the illusion of “particles” leaving the system, whereas in (b) a positive error in $\eta$ gives the illusion of “particles” entering the system.

Figure 5

Frequency histogram of $\eta ∕\alpha$ for the case when $\Phi (x,y)=0$. The histogram is constructed from 100 equally spaced isointensity surfaces using Eq. (14). The vertical axis is the number of occurrences of $\eta ∕\alpha$ within the interval $\Delta (\eta ∕\alpha )={10}^{-3}$, whereas the horizontal axis shows the various retrieved values of $\eta ∕\alpha$. The value $\eta ∕\alpha =2.283$ at the peak of the histogram is taken as the initial guessed value of $\eta ∕\alpha$ for our diffusion relaxation scheme.

Figure 6

The evolution of $\eta ,\alpha ,N$, and $X$ in the diffusion relaxation numerical iteration scheme for a typical simulation $(⟨\mid \Phi \mid ⟩=0.49)$. (a) $\eta$ converges to a precise value in the long-term evolution. From above ${\eta }_{+}\to 2.000443$, from below ${\eta }_{-}\to 2.000439$. The exact value is $\eta =2$. (b) When $\eta$ is overestimated, $\alpha$ was initially underestimated (its exact value is unity); however, it subsequently evolves towards ${\alpha }_{-}\to 1.007$. When $\eta$ is underestimated, $\alpha$ is relatively constant, illustrating that $\alpha$ is less sensitive to a negative error in $\eta$ when compared to a positive error in this quantity. (c) shows that the normal phase gradient along the boundary tends towards zero from both sides, whereas (d) illustrates the fractional variation of $\eta$ as it converges towards a solution.

Figure 7

A typical frequency histogram of $\alpha$ at the end of the diffusion relaxation scheme, for $\alpha =1.01\pm 0.04$. The error is taken as the FWHM of the histogram. The histogram is constructed from 100 equally spaced isointensity surfaces using Eq. (6). The vertical axis is the number of occurrences of $\alpha$ within the interval $\Delta \alpha ={10}^{-3}$, whereas the horizontal axis shows the various retrieved values of $\alpha$.

Figure 8

(a) Relative error in the retrieved phase taken at the end of the diffusion relaxation scheme. This relative error is consistent with an error of $\eta$ being positive and also consistent with the accuracy with which the phase can be retrieved. (b) and (c) show that the rms error in the phase and the phase gradient quickly decays with increasing iteration number $k$.

Figure 9

Comparison between the exact function $f\left(I\right)=100\sin \left(\pi I\right)$ (upper curve) and the inferred function (lower curve), which can be fitted with $-131.14+100.59\sin \left(\pi I\right)$. This illustrates that the nonlinear function $f\left(I\right)$ can be recovered accurately. The constant offset is indicative of the fact that the wave-function phase can only be retrieved up to an arbitrary constant.

Figure 10

Schematic illustration of the iteration scheme to infer the TDCGL equation. Multiple modulated plane-wave states $P\left(I_q\right)$ and a localized state $L\left(I\right)$ are prepared. An initial guess of $\tilde{\eta }∕\alpha$ is obtained. The dissipation on the localized state $g_L(x,y)$ is obtained from the dissipation on the plane-wave states $g\left(I\right)$ (note that the measurements are in units of $\alpha$). Given the guess of $\tilde{\eta }∕\alpha$ and the approximation for the dissipation, the diffusion relaxation (DR) scheme is used to find $\tilde{\Phi },\alpha$, and an improved value of $\eta ∕\alpha$. The numerical scheme is repeated if this improved value is not the required stationary value.

Figure 11

(a) Dotted points are the measured dissipation and continuous curves are the exact dissipation. These plots show that the dissipation can be inferred very accurately. (b) The error in the dissipation function, $g\left(I\right)-g_E\left(I\right)$ [where $g\left(I\right)$ is the measured dissipation and $g_E\left(I\right)$ is the exact dissipation], for the case $g_E\left(I\right)=10I$. The top curve is the error in the measured dissipation in the first measurement, and the bottom curve is the error at the third measurement. The lower error at the third measurement illustrates convergence of the numerical dissipation measurement scheme.

Figure 12

(a) Measured $f\left(I\right)$ (bottom) compared to the exact $f_E\left(I\right)=100\sin \left(\pi I\right)$ [for the case $g_E\left(I\right)=10I$]. The measured $f\left(I\right)$ can be fitted with a function $f\left(I\right)=-132.45+100.45\sin \left(\pi I\right)$, and the small error (neglecting the constant offset $-132.45$) indicates accurate measurement of the function. (b) The error in measuring $f\left(I\right)$; most errors are within 2% and are expected to decrease if simulations are performed at higher spatial resolution.

Figure 13

Schematic illustration of the trajectories $T_1$ and $T_2$ corresponding to a path in $I_1$ and $I_2$ space. (a) $T_1$ traverses a path in physical space corresponding to an isointensity surface in $I_1$. (b) Along the same trajectory in physical space, the intensity $I_2$ varies in $T_2$, with the end points identified. In general, the trajectory $T_2$ is not an isointensity surface; however, there are pairs of points with the same intensity.