SPIDER Simulator — Spectral Phase Interferometry for Direct Electric-field Reconstruction

Interactive simulation of temporal characterization of femtosecond laser pulses. Move any slider: the SPIDER interferogram, the FTSI reconstruction, and all derived quantities are recomputed and redrawn in real time.

Test pulse

Center wavelength λ0 (nm)
Bandwidth FWHM Δλ (nm)
GDD φ2 (fs2)
TOD φ3 (fs3)
FOD φ4 (fs4)
Phase ripple ampl. (rad)
Phase ripple period (fs)

SPIDER apparatus

Replica delay τ (fs)
Spectral shear Ω (rad/fs)
Sideband filter HWHM (fs)
Detector noise (% rms)

Spectrogram view

Gate FWHM (fs)

SPIDER setup schematic

input pulse E(t) BS etalon / interferometer pulse pair, delay τ grating stretcher strong chirp (GDD₁ = τ/Ω) SFG in χ⁽²⁾ crystal sheared replicas spectrometer interferogram S(ω) FTSI algorithm filter, unwrap, concatenate → φ(ω)
One replica pair (delay τ) is sum-frequency mixed with a strongly chirped copy of the pulse. Each replica overlaps a different quasi-monochromatic slice of the chirped pulse, so the two upconverted replicas are relatively shifted in frequency by the spectral shear Ω = τ/GDDstretcher. The spectrometer records their interference. All spectra below are shown at the fundamental frequency (the constant SFG upconversion offset is dropped for clarity).

1. Fundamental spectrum and spectral phase

Measured spectral intensity (filled), true spectral phase (blue) and SPIDER-reconstructed phase (red dashed, right axis). Constant and linear phase (absolute phase and arrival time) are not observables and are removed before comparison.

2. Temporal intensity

True pulse |E(t)|2 (blue), reconstruction from the SPIDER phase (red dashed) and transform-limited pulse of the same spectrum (gray). Green: instantaneous frequency detuning Δω(t) of the true pulse (right axis), shown where the intensity exceeds 2% of peak.

3. SPIDER interferogram S(ω)

Simulated spectrometer signal S(ω) = I(ω) + I(ω−Ω) + 2√I(ω)I(ω−Ω) cos[φ(ω)−φ(ω−Ω)+ωτ], including detector noise. The fringe spacing is ≈2π/τ; deviations from perfectly periodic fringes encode the spectral phase gradient.

4. Fourier transform of the interferogram (FTSI)

|C(t)| of the interferogram in pseudo-time (log scale). The DC peak at t = 0 carries only the spectra; the AC sidebands at t = ±τ carry the phase information. The green band is the super-Gaussian filter that isolates the +τ sideband. Sidebands must be well separated from DC.

5. Extracted phase difference θ(ω) = φ(ω) − φ(ω−Ω)

Phase of the filtered sideband after removing the calibration term ωτ (red), compared with the exact finite difference of the true phase (blue). This is the raw quantity SPIDER measures; the spectral phase follows by concatenation in steps of Ω.

6. Group delay Tg(ω) = dφ/dω

Group delay across the spectrum, true (blue) vs reconstructed (red dashed). A linear slope corresponds to GDD, curvature to TOD. This is often the most physically readable form of the spectral phase.

7. Reconstruction error

Pointwise phase error φrec(ω) − φtrue(ω) after removing the best-fit constant and linear term (red), with the spectral intensity as backdrop (right axis). Errors in the spectral wings matter little because the intensity there is negligible.

8. Spectrogram (Gabor / gated FT)

Time-frequency map |∫E(t')g(t'−t)e−iωt'dt'|2 with a Gaussian gate, for the true or reconstructed pulse (toggle in the sidebar). The dashed black curve is the true group delay Tg(ω): for a well-characterized pulse the ridge follows it. Chirp appears as a tilt, TOD as curvature.
SPIDER algorithm, step by step (what this page computes)
  1. Test pulse. A Gaussian spectrum centered at λ0 with intensity-FWHM Δλ gets the Taylor phase φ(ω) = φ2x2/2 + φ3x3/6 + φ4x4/24 + a·sin(xT), with x = ω−ω0.
  2. Interferogram. Two replicas separated by τ are upconverted with a chirped pulse, shearing one by Ω. The recorded signal is S(ω) = I(ω) + I(ω−Ω) + 2√(I(ω)I(ω−Ω)) cos[φ(ω)−φ(ω−Ω)+ωτ], plus multiplicative and additive detector noise.
  3. Fourier filtering (Takeda FTSI). S(ω) is Fourier transformed to pseudo-time. The sideband at t = +τ is isolated with a super-Gaussian window (order 6) and transformed back, giving the complex signal A(ω)A*(ω−Ω)eiωτ.
  4. Calibration. Its unwrapped argument minus the known linear term ωτ (obtained in practice from a separate unsheared calibration trace) yields θ(ω) = φ(ω) − φ(ω−Ω).
  5. Concatenation. Starting at the spectral peak, φ(ωk+Ω) = φ(ωk) + θ(ωk+Ω) builds the phase on a grid with spacing Ω; linear interpolation fills the spectrometer grid. (Alternative: direct integration φ(ω) = ∫θ/Ω dω.)
  6. Field reconstruction. The reconstructed phase is combined with the independently measured spectrum √I(ω) and inverse Fourier transformed to give E(t). SPIDER is direct and non-iterative: no retrieval loop is needed.
  7. Design rules visible in this simulator: τ must be large enough to separate the sidebands from DC but small enough that the spectrometer resolves the fringes (≥4 samples per fringe); Ω must be small enough to sample the phase structure (Whittaker-Shannon: features of period Tmod require Ω < π/Tmod) but large enough for a good signal-to-noise on θ.