Wave Dispersion Calculator
Use Cauchy coefficients to model wavelength dependent refractive behavior in transparent chemical media.
Plotly Dispersion Graph
The chart compares refractive index with phase and group velocity across your selected wavelength range.
Formula Used
Cauchy Dispersion Equation
n(λ) = A + B/λ² + C/λ⁴
First Derivative
dn/dλ = -2B/λ³ - 4C/λ⁵
Group Index
ng = n - λ(dn/dλ)
Velocities and Delays
vp = c/n, vg = c/ng, delay = length/velocity
This page uses wavelength in micrometers inside the Cauchy model, while user inputs remain in nanometers for easier laboratory interpretation.
How to Use This Calculator
- Enter a medium name for reporting clarity.
- Provide Cauchy coefficients A, B, and C.
- Enter the central wavelength in nanometers.
- Set spectral width for pulse spread estimation.
- Enter path length through the chemical medium.
- Define a wavelength range for the Plotly graph.
- Click Calculate Dispersion.
- Review results, graph trends, and export files.
Example Data Table
Illustrative example using A = 1.3220, B = 0.0030, C = 0.0002, path length = 10 cm, spectral width = 40 nm.
| Wavelength (nm) | Refractive Index | Group Index | Phase Velocity (m/s) | Group Velocity (m/s) | Group Delay (ns) |
|---|---|---|---|---|---|
| 450 | 1.34169212 | 1.39083097 | 2.234e+8 | 2.155e+8 | 0.46393127 |
| 550 | 1.33410300 | 1.36268028 | 2.247e+8 | 2.200e+8 | 0.45454122 |
| 650 | 1.33022100 | 1.34890382 | 2.254e+8 | 2.222e+8 | 0.44994588 |
Frequently Asked Questions
1. What does this calculator estimate?
It estimates refractive index, group index, phase velocity, group velocity, delays, wave number, and approximate pulse spreading across a selected wavelength band.
2. Why use Cauchy coefficients?
Cauchy coefficients model transparent media well over limited spectral ranges. They are useful for many chemistry, optics, and spectroscopy calculations involving visible wavelengths.
3. Is this suitable for strongly absorbing media?
No. Strong absorption often requires more advanced dispersion models. This tool works best for transparent or weakly absorbing media over practical wavelength windows.
4. What does group index mean?
Group index describes how a pulse envelope travels through a medium. It is the main value behind group velocity and pulse delay estimates.
5. Why are phase and group velocities different?
Phase velocity tracks a single sinusoidal phase front. Group velocity tracks a pulse or packet. Dispersion causes them to differ when refractive index changes with wavelength.
6. What is the pulse spread estimate?
It is the approximate difference in group delay between the low and high edges of the chosen spectral width over the specified path length.
7. Which units does the calculator use?
Inputs use nanometers and centimeters. Internal Cauchy calculations use micrometers. Results are shown in m/s, ns, ps, THz, and rad/m where relevant.
8. Can I use literature coefficients directly?
Yes, provided the coefficients match the same Cauchy form and wavelength unit convention. Always confirm whether the source expects micrometers or another unit.