Group Delay / GDD

The GD/GDD window computes the spectral phase of a coating and its derivatives with respect to angular frequency: group delay (GD), group-delay dispersion (GDD), and third-order dispersion (TOD). These quantities describe how a coating delays different parts of an optical pulse.

The phase comes from the complex reflection or transmission coefficient:

φ(ω) = arg(r)  or  arg(t)
GD  = -dφ/dω       [fs]
GDD = -d²φ/dω²     [fs²]
TOD = -d³φ/dω³     [fs³]

Settings

Quantity: phase φ, GD, GDD, or TOD.

Reflection / Transmission: take the phase from the reflected or transmitted complex amplitude.

Polarization: the average of s and p, s, or p. The average uses the same per-polarization arithmetic mean as the matching merit operand.

Side: evaluate the front coating or the back coating.

Wavelength range: the span plotted and exported, in nm. TFStudio chooses the sampling automatically and adds local samples around pronounced reflection or transmission minima. There is no derivative or sampling step to tune.

AOI: angle of incidence in degrees, measured in the incident medium.

Reference wavelength: shifts the displayed phase to zero at the selected wavelength. This constant offset does not change GD, GDD, or TOD.

Targets: shows enabled GD, GDD, or TOD merit-function targets that match the selected reflection or transmission response, polarization, and AOI. Point operands appear as X markers. Flatness operands show their target level and wavelength band. Phase targets are not overlaid because the displayed phase may have an arbitrary reference offset. Current phase-dispersion merit operands evaluate the front coating normally and the back coating for a back-only design, so their overlays appear only on the side they score.

How the values are calculated

GD, GDD, and TOD are evaluated point by point through third-order Taylor arithmetic in the characteristic matrix. The derivatives come from the complex logarithmic derivative of r or t; phase unwrapping is used only to draw the phase curve. TFStudio uses n + ik with an exp(-iωt) time factor, then applies the conjugate-Macleod convention once so a material transit time is positive, with the same sign as the Material Dispersion window.

Formula materials are differentiated exactly. A tabulated material gives the exact derivative of its shape-preserving PCHIP curve. PCHIP is C1: GD is continuous, while higher derivatives can show finite steps at table knots and TOD is especially sensitive to how sparse data is represented. For coating reflection and transmission, both tabulated n and k contribute to this continuity limit. GDD and TOD plots leave gaps at their knot jumps, and the warning badge on the control row names the table models involved. A saved smooth fit replaces the table only inside its stated validity range and is named there too. Wavelengths outside any material model range are left blank with a reason instead of treating a clamped endpoint as non-dispersive data.

How to read it

For a chirped mirror, GD should follow the target ramp across the band and GDD should hold the intended value used for pulse compensation. A narrow positive or negative GD feature beside a reflection zero is expected phase behavior. Read it with the coefficient magnitude: little reflected energy occupies a deep reflectance minimum, although the same feature can matter when the coating is used in transmission.

The data table lists phase and all three derivatives against wavelength for export.

References

• H. A. Macleod, Thin-Film Optical Filters, 5th ed., Ch. 11, Eq. 11.17.
• J. Birge and F. X. Kärtner, “Efficient analytic computation of higher-order dispersion from optical interferometers,” Applied Optics 45, 1478-1483 (2006), doi:10.1364/AO.45.001478.
• S. Diddams and J.-C. Diels, Journal of the Optical Society of America B 13, 1120 (1996).