FibDisp — About & Model

Purpose and scope

FibDisp simulates the propagation of an ultrashort laser pulse in a single-mode gas-filled hollow-core fiber or capillary. Propagation is calculated with a symmetric split-step Fourier method applied to a generalized nonlinear Schrödinger equation containing second- and third-order dispersion, linear attenuation, Kerr self-phase modulation, self-steepening, and input quadratic and cubic spectral phase.

The program provides Gaussian, hyperbolic-secant and super-Gaussian input pulses; calculated gas/waveguide coefficients and a Manual mode; selectable propagation terms; two self-steepening solvers; temporal and spectral evolution maps; post-fiber GDD display; spectral-phase fitting; GDD-compressor analysis; and one-parameter sweeps.

The web application performs the numerical calculation locally in the browser. The numerical model itself is the same FibDisp physics core used by the desktop application.

Coordinates and principal quantities

Symbol	Definition
\(z\)	Propagation coordinate along the fiber; \(L\) is the total fiber length.
\(t\)	Laboratory time.
\(T\)	Retarded time, \(T=t-\beta_1z\).
\(U(z,T)\)	Dimensionless complex pulse envelope used by the numerical model.
\(A(z,T)\)	Power-normalized envelope, \(A=\sqrt{P_0}U\), in \(\sqrt{\mathrm W}\).
\(P(z,T)\)	Instantaneous pulse power, \(P=P_0\|U\|^2\), in W.
\(E_p\)	Pulse energy, \(E_p=\int P(T)\,dT\).
\(\lambda_0\)	Carrier wavelength; \(\nu_0=c/\lambda_0\), \(\omega_0=2\pi\nu_0\).
\(f_{\rm FFT}\)	Centred numerical FFT frequency coordinate.
\(\nu\)	Physical optical frequency, \(\nu=\nu_0-f_{\rm FFT}\).
\(\Omega\)	Physical angular-frequency detuning, \(\Omega=\omega-\omega_0=-2\pi f_{\rm FFT}\).
\(\beta_2,\beta_3\)	Second- and third-order derivatives of the modal propagation constant with respect to angular frequency at \(\omega_0\).
\(\alpha\)	Power attenuation coefficient in m\(^{-1}\).
\(\gamma\)	Kerr nonlinear coefficient in W\(^{-1}\)m\(^{-1}\).
\(R\)	Capillary radius; \(A_{\rm eff}\) is the effective modal area.

Fourier and sign convention

\[ \widetilde U(f_{\rm FFT})=\int_{-\infty}^{+\infty}U(T)e^{-i2\pi f_{\rm FFT}T}\,dT, \qquad \frac{\partial}{\partial T}\longleftrightarrow i2\pi f_{\rm FFT}. \]

\[ \boxed{\Omega=-2\pi f_{\rm FFT}},\qquad \boxed{\nu=\nu_0-f_{\rm FFT}},\qquad \lambda=\frac{c}{\nu}. \]

Propagation model

Linear spectral operator

In retarded time \(T=t-\beta_1z\), the linear spectral contribution is

\[ \left.\frac{\partial\widetilde U}{\partial z}\right|_{\rm L}=D(\Omega)\widetilde U, \qquad D(\Omega)=-\frac{\alpha(\omega_0+\Omega)}{2} +i\frac{\beta_2}{2}\Omega^2+i\frac{\beta_3}{6}\Omega^3. \]

For Manual mode, \(\alpha\) is constant. For a built-in gas, the frequency-dependent capillary-loss function is used.

Nonlinear envelope equation

\[ A(z,T)=\sqrt{P_0}\,U(z,T),\qquad P(z,T)=P_0|U(z,T)|^2, \]

\[ \boxed{ \left.\frac{\partial U}{\partial z}\right|_{\rm NL} =i\gamma P_0|U|^2U -\frac{\gamma P_0}{\omega_0}\frac{\partial}{\partial T}\left(|U|^2U\right) }. \]

The first nonlinear term is self-phase modulation. The derivative term is self-steepening (optical-shock correction).

Combined implemented equation

\[ \boxed{ \frac{\partial U}{\partial z} =\mathcal F^{-1}\!\left\{D(\Omega)\widetilde U\right\} +i\gamma P_0|U|^2U -\frac{\gamma P_0}{\omega_0}\frac{\partial}{\partial T}\left(|U|^2U\right) }. \]

Gas and capillary model

For the built-in gases, FibDisp evaluates the gas refractive index with pressure scaling and combines it with the large-core capillary approximation. The resulting modal index is differentiated numerically around the carrier to obtain the displayed \(\beta_2\) and \(\beta_3\). The nonlinear coefficient is

\[ \gamma=\frac{n_2\omega_0}{cA_{\rm eff}}. \]

Manual mode allows direct specification of \(\beta_2\), \(\beta_3\), \(\alpha\), and \(\gamma\).

Numerical propagation

The full propagation uses a symmetric split-step Fourier method. Each global propagation step applies a half linear step in the spectral domain, a nonlinear step in retarded time, and a second half linear step.

For self-steepening, two nonlinear solvers are available:

Fast exponential: a fast frozen nonlinear-step approximation.
Accurate RK4 (adaptive): integrates the nonlinear equation directly with RK4 and automatically subdivides the nonlinear step when required by nonlinear phase or self-steepening strength.

Boundary warnings indicate when appreciable field amplitude reaches the temporal or spectral edge of the computational window.

Phase and compressor conventions

\[ \phi_{\rm GDD}(\Omega)=\frac{\mathrm{GDD}}{2}\Omega^2, \qquad \phi_{\rm TOD}(\Omega)=\frac{\mathrm{TOD}}{6}\Omega^3. \]

The displayed temporal instantaneous-frequency shift is

\[ \Delta\nu(T)=-\frac{1}{2\pi}\frac{d\phi_T}{dT}, \]

and the spectral group delay is \(GD=d\phi/d\omega\). A predominantly positive output quadratic spectral phase is compensated by a negative applied GDD, and vice versa.

Phase Analysis performs a spectral-power-weighted polynomial fit over the selected physical spectral interval. GDD Compressor Design applies a pure quadratic spectral phase and can optimize it according to the selected temporal metric.

Application workflow

Choose gas, pressure, radius, length, pulse shape, energy, TL duration, wavelength and input phase in Settings.
Inspect \(\beta_2\), \(\beta_3\), \(\alpha\), and \(\gamma\), then preview the input field.
Select propagation terms and the self-steepening solver in Start & Results.
Run the propagation and check boundary warnings.
Inspect temporal/spectral output, transform limit, chirp, group delay, and propagation maps.
Use Phase Analysis and GDD Compressor Design when phase or compression is of interest.
Use Parameter Sweep to study trends while keeping track of the baseline settings.
Repeat selected calculations with finer numerical grids before interpreting small differences quantitatively.

Model limitations

The implemented propagation model includes Kerr nonlinearity, self-steepening, loss, and dispersion through \(\beta_3\). It does not explicitly include:

photoionization and plasma generation;
ionization loss;
a time-resolved molecular Raman response;
dispersion coefficients beyond \(\beta_3\) in the propagation Taylor expansion;
propagation with the complete frequency-dependent modal \(\beta(\omega)\);
multimode transverse dynamics;
longitudinally varying capillary radius or pressure profiles;
real-compressor amplitude response or higher-order compressor phase beyond the applied pure quadratic phase.

The large-core capillary approximation is used for the built-in gas coefficients. Far from the carrier, particularly when a spectrum spans a very large fraction of an octave, the local \(\beta_2+\beta_3\) Taylor expansion can become less accurate than propagation based on the full modal dispersion relation. In regimes where omitted physics is important, the result should be interpreted as the prediction of the model defined above, not as a complete description of every experimental process.

References

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Applied Physics Letters 68, 2793–2795 (1996). DOI: 10.1063/1.116609.
M. Nisoli et al., “Compression of high-energy laser pulses below 5 fs,” Optics Letters 22, 522–524 (1997).
C. Vozzi et al., “Optimal spectral broadening in hollow-fiber compressor systems,” Applied Physics B 80, 285–289 (2005). DOI: 10.1007/s00340-004-1721-1.
B. Schenkel et al., “Generation of 3.8-fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,” Optics Letters 28, 1987–1989 (2003).