Theory

Dispersion Theory

Phase, group delay, GDD, and differential group delay from thin-film transfer matrices

This chapter introduces the physical quantities behind the four dispersion detectors: Phase φ(λ), Group Delay (GD), Group Delay Dispersion (GDD), and Differential Group Delay (DGD). These quantities are derived from the complex-valued transmission and reflection coefficients produced by the transfer matrix method.

Complex Phase from Transfer Matrix Coefficients

The transfer matrix method yields complex reflection and transmission coefficients for each polarization state (S and P):

r_s(\lambda),\; r_p(\lambda),\; t_s(\lambda),\; t_p(\lambda)

Each coefficient is a complex number. The spectral phase is defined as:

\varphi(\lambda) = \arg\bigl[c(\lambda)\bigr]

where $c(\lambda)$ is one of the four coefficients above.

Because $\arg$ returns values in $(-\pi, \pi]$ , the raw phase often contains $2\pi$ discontinuities. The application provides an optional phase unwrapping step that removes these jumps and produces a continuous phase curve.

Group Delay

Group Delay is the negative first derivative of the spectral phase with respect to angular frequency:

\text{GD}(\omega) = -\frac{d\varphi}{d\omega}

Since the application works in wavelength space, the conversion uses:

\omega = \frac{2\pi c}{\lambda}, \quad d\omega = -\frac{2\pi c}{\lambda^2}\,d\lambda

The sign convention follows standard optics usage: a positive GD indicates a time delay. The unit is femtoseconds (fs).

Physically, GD represents the envelope delay of a narrow-band pulse centered at a given wavelength. A flat GD across a wavelength band means all spectral components experience the same delay, preserving pulse shape.

Group Delay Dispersion

GDD is the derivative of group delay with respect to angular frequency, equivalently the second derivative of phase:

\text{GDD}(\omega) = \frac{d\,\text{GD}}{d\omega} = \frac{d^2\varphi}{d\omega^2}

The unit is fs² (femtosecond-squared).

GDD sign	Effect on pulse
Positive	Normal (positive) chirp: longer wavelengths lead, shorter wavelengths lag
Negative	Anomalous (negative) chirp: shorter wavelengths lead
Zero	No pulse broadening from second-order dispersion

GDD is the primary figure of merit for dispersive mirror design (e.g., chirped mirrors for ultrafast lasers). A GDD target curve can be used as an optimization objective.

Differential Group Delay

DGD quantifies the polarization-dependent time delay:

\text{DGD}(\lambda) = \text{GD}_P(\lambda) - \text{GD}_S(\lambda)

The unit is femtoseconds (fs). DGD is a direct measure of polarization-mode dispersion (PMD).

In an isotropic multilayer at normal incidence, S and P responses are identical and DGD is zero. At oblique incidence, S and P experience different effective optical paths through the stack, producing nonzero DGD. This effect increases with incidence angle and with the number of layer interfaces.

DGD is relevant for:

telecommunications coatings where polarization-dependent delay causes signal distortion,
oblique-incidence filter design where S-P splitting must be minimized.

Numerical Differentiation and Confidence Metadata

GD, GDD, and DGD are obtained by numerical differentiation of the phase. The application uses finite-difference methods on an internal sampling grid.

Two consequences arise:

Edge artifacts: near the boundaries of the wavelength range, the finite-difference stencil lacks neighboring points on one side. The solver marks these edge regions as low-confidence and reports a trusted wavelength range in the result metadata. Points outside this range are omitted from charts and tables.
Sampling density: the internal grid density is controlled by the Numerical Quality setting on the Optics page. Low uses fewer internal points (faster, noisier), while High uses a denser grid (slower, more stable). This internal grid is independent of the user-specified wavelength step.

To maximize the trusted wavelength range, set the sweep wavelength range wider than the region of interest. The edge margin that is trimmed depends on the quality setting and the local spectral complexity.

Connection to Pulse Behavior

Quantity	Pulse-domain interpretation
Phase φ(λ)	Spectral phase profile; controls the instantaneous frequency structure of the output pulse
GD	Envelope arrival time; a wavelength-dependent GD means different spectral components arrive at different times
GDD	Pulse broadening rate; dominant source of temporal broadening for transform-limited pulses
DGD	Polarization-dependent arrival time; relevant when the input pulse has mixed polarization

For ultrafast optics, GDD is typically the first quantity to examine. For telecom filter design, DGD is often more important.

For setup and result-page usage, see Optical Parameters — Dispersion Detectors and Dispersion Results.

Depth-Resolved Quantities

Energy flux, absorption, and electric fields versus depth

Structure Configuration

Structural modeling, layer stacks, and validation