Kramers–Kronig Calculator

Calculator Inputs

Example Data Table

Formula Used

For a causal linear response (susceptibility-like) function χ(ω) with ω > 0, one common Kramers–Kronig pair is:

• Re(χ)(ω₀) = (2/π) P ∫₀^∞ [ ω Im(χ)(ω) / (ω² − ω₀²) ] dω
• Im(χ)(ω₀) = (−2ω₀/π) P ∫₀^∞ [ Re(χ)(ω) / (ω² − ω₀²) ] dω

This calculator approximates the principal-value integral using discrete data, midpoint-style weights, and an exclusion band around each ω₀ to handle the pole numerically.

How to Use

1. Choose whether you are converting Imaginary → Real or Real → Imaginary.
2. Paste your spectrum as lines of ω, value (at least three rows).
3. Set an exclusion fraction to control principal-value handling near ω₀.
4. Optionally set epsilon to stabilize calculations near sharp resonances.
5. Click Submit to see the transformed component above the form.
6. Use the download buttons to export the full computed table.

Technical Article

1) Why Kramers–Kronig Relations Matter

Kramers–Kronig relations connect dispersion and absorption in any linear, causal medium. If you measure only one component of a response function, the other component is constrained by physics. This is widely used in optics, acoustics, electronics, and material characterization workflows.

2) Causality, Analyticity, and Practical Data

Causality implies the frequency-domain response is analytic in the upper half of the complex plane, leading to Hilbert-transform style integrals. Real measurements are finite, noisy, and sampled on a grid, so numerical strategies are required to approximate the principal-value integral reliably.

3) Common Physics Inputs and Outputs

In optical spectroscopy, users often transform the imaginary part of susceptibility or dielectric response into its real part to recover dispersion. With refractive index data, a related workflow links n(ω) and k(ω). In electronics, impedance and admittance components can be cross-checked for consistency using Kramers–Kronig constraints.

4) Frequency Grid and Sampling Guidance

A smooth, strictly increasing frequency array improves stability. Dense sampling near resonances captures steep features; coarser sampling is usually acceptable in slowly varying tails. If your data spacing changes abruptly, transform results may show artificial ripples. Resampling onto a smoother grid can help.

5) Handling the Principal Value Numerically

The integrand has a pole at ω = ω₀, so the integral is interpreted as a principal value. This calculator excludes points within a configurable fraction of ω₀ and applies midpoint-style weights. The exclusion fraction is a practical knob for suppressing near-pole spikes.

6) Bandwidth, Edge Effects, and Extrapolation

Finite measurement bandwidth is the dominant source of error. Missing low-frequency or high-frequency tails can bias the baseline and distort dispersion at the edges. In professional analysis, analysts extend spectra with physically motivated extrapolations (power-law or Drude/Lorentz tails) before applying Kramers–Kronig transforms.

7) Validation Checks You Should Run

Compare transformed results against known limits: causality-consistent responses should be smooth and free of nonphysical oscillations. If you transform Imaginary → Real and then back Real → Imaginary on the same grid, the recovered curve should broadly match the input within expected numerical error and bandwidth limits.

8) Reporting, Export, and Traceability

For audits and publications, record your units, grid range, exclusion settings, and any smoothing or extrapolation steps. Exported CSV supports plotting in scientific tools, while the PDF provides a compact snapshot of parameters and the first rows of results for documentation. Consistent reporting makes your dispersion analysis reproducible.

FAQs

1) What type of data should I paste?

Paste two columns per line: frequency ω and the component you measured (real or imaginary). Use at least three rows, and keep ω strictly increasing after sorting.

2) Why do results look unstable near a resonance?

Near sharp features, the integrand changes rapidly and approaches a pole. Increase sampling density, raise the exclusion fraction slightly, or add a small epsilon to reduce numerical spikes.

3) Does the frequency unit selection affect calculations?

No. Units are displayed for clarity only. The transform uses the numeric ω values you provide, so keep your dataset internally consistent when switching between Hz, rad/s, or eV.

4) What does the exclusion fraction mean?

It defines a small band around each ω₀ that is skipped to approximate the principal value. Larger values suppress pole artifacts but can smooth real physical structure if set too high.

5) When should I use epsilon?

Use epsilon if the output shows extreme spikes due to near-zero denominators. Start very small and increase gradually. Epsilon is a stabilizer, not a replacement for better bandwidth and sampling.

6) Why are edge regions often less reliable?

Kramers–Kronig integrals depend on the full spectrum from zero to infinity. Limited bandwidth forces truncation, so the beginning and end of your frequency range often carry the largest systematic error.

7) Can I use this for n and k directly?

This tool implements susceptibility-style relations on ω-grid data. For refractive index workflows, first convert to a consistent response form (such as dielectric function) or use a dedicated n–k Kramers–Kronig model.