Four Wave Mixing Calculator

Inputs

Choose a mode, enter pump and signal, then compute idler and phase matching.

Input mode

Wavelengths are converted internally to frequencies.

Degenerate pumps

Use pump 1 value for pump 2

Useful for 2f_p − f_s cases.

Interaction length L (m)

Used for coherence and sinc² calculations.

Pump 1 (nm)

Typical telecom example: 1550 nm or 193.5 THz.

Pump 2 (nm)

Disable by enabling degenerate pumps.

Signal (nm)

Idler is computed from frequency mixing.

Medium and phase matching

Set refractive indices for each wave, or use one shared value.

Use one refractive index

When enabled, all indices are set to n.

Common index n

Example for silica near 1550 nm: ~1.44.

n (pump 1)

n (pump 2)

n (signal)

n (idler) (optional)

If blank, idler index is approximated as n(signal).

Optional efficiency estimate

Provide γ, pump powers, and attenuation to estimate relative conversion.

Nonlinear parameter γ (1/W/m)

Common in waveguide and fiber modeling.

Pump 1 power (W)

Pump 2 power (W)

Attenuation α (1/m)

Used for effective length: (1 − e^−αL)/α.

Four wave mixing follows frequency conservation for the generated idler: f_i = f_p1 + f_p2 − f_s. When pumps are degenerate, this becomes f_i = 2f_p − f_s.

Phase matching is evaluated using the wavevector mismatch: Δk = k_p1 + k_p2 − k_s − k_i, with k = 2πn/λ. The coherence length is L_c = π/|Δk|, and the acceptance factor is sinc²(Δk·L/2).

For an optional relative efficiency estimate, the calculator uses η ∝ γ² P₁P₂ L_eff² sinc²(Δk·L/2), where L_eff = (1 − e^−αL)/α when α > 0.

Select wavelength or frequency mode based on your data.
Enter pump 1, pump 2, and signal values with the shown units.
Set interaction length L to match your medium or device.
Choose a single refractive index, or enter separate indices.
Optionally enter γ, pump powers, and attenuation for an estimate.

Mode	Pump 1	Pump 2	Signal	L (m)	n	Computed Idler (approx.)
Wavelength	1550 nm	1550 nm	1545 nm	0.50	1.44	≈ 1555 nm
Frequency	193.5 THz	193.5 THz	194.1 THz	0.20	1.44	≈ 192.9 THz

The example values are illustrative. Real outcomes depend on dispersion and accurate refractive indices.

1) What four wave mixing does

Four wave mixing (FWM) is a nonlinear optical process where interacting waves exchange energy through the medium’s third-order response. Two pump waves and a signal can generate an idler whose frequency satisfies the mixing relation used in this calculator. FWM supports wavelength conversion, parametric amplification, and spectral broadening in fibers, waveguides, and resonators.

2) Energy conservation in practical terms

In frequency units, the idler is computed as f_i = f_p1 + f_p2 − f_s. When the pumps are degenerate, the relation becomes 2f_p − f_s. This is why a small signal detuning around the pump produces an idler mirrored on the opposite side, a common feature in wavelength conversion experiments.

3) Why phase matching matters

Even with perfect energy conservation, FWM efficiency depends strongly on phase matching. The calculator evaluates Δk from the wavevectors of the interacting fields. When Δk is near zero, the nonlinear polarization adds constructively along the device and conversion grows rapidly. As Δk increases, contributions cancel and the sinc² term reduces output.

4) Coherence length and acceptance bandwidth

The coherence length L_c = π/|Δk| is a useful design metric. If the physical interaction length is much greater than L_c, the process oscillates and the net conversion can flatten. Shorter devices may tolerate larger mismatch, while longer devices require tighter dispersion control.

5) Role of refractive index and dispersion

Accurate refractive indices are essential because k = 2πn/λ. In real materials, n varies with wavelength, so the pumps, signal, and idler may each experience different phase velocities. Using separate indices in this calculator gives a quick way to test how dispersion shifts Δk and L_c.

6) Interpreting the efficiency estimate

The optional relative efficiency uses η ∝ γ²P₁P₂L_eff²sinc²(ΔkL/2). Here γ summarizes nonlinear strength and mode confinement. The estimate is best for comparisons: try different lengths, losses, or pump powers to see trends before running a full coupled-mode simulation.

7) Losses and effective length

Propagation loss reduces the usable interaction region, which is captured by L_eff. When attenuation α is small, L_eff approaches L. When α is larger, L_eff saturates and additional physical length yields diminishing returns. This helps explain why low-loss platforms can outperform shorter, higher-loss devices.

8) Common use cases

Engineers use FWM for wavelength routing in telecom links, idler generation in spectroscopy, parametric gain in integrated photonics, and frequency translation for sensing systems. This calculator packages the core relationships into a fast workflow, with CSV and PDF exports for lab notes, design reviews, and documentation.

1) What is the idler in four wave mixing?

The idler is a newly generated wave whose frequency is set by the pumps and signal through the mixing relation. It often appears symmetrically around the pump in frequency space for degenerate pumping.

2) Why can my idler be far from the signal wavelength?

Frequency mixing is linear in frequency, not wavelength. When you convert back to wavelength, the mapping is nonlinear, so equal frequency shifts do not translate to equal wavelength shifts.

3) What does Δk represent?

Δk is the wavevector mismatch among the interacting waves. If Δk is near zero, contributions add coherently along the device. Larger mismatch causes cancellation and reduces conversion efficiency.

4) How should I choose refractive indices?

Use wavelength-dependent indices when possible, especially in dispersive media. If you only have one value, start with a common index to estimate trends, then refine with separate indices for better accuracy.

5) What is coherence length used for?

Coherence length indicates how far the process builds constructively before phase slippage causes cancellation. If your device length greatly exceeds the coherence length, conversion can oscillate and average out.

6) Is the efficiency number an absolute conversion efficiency?

No. It is a relative estimate based on simplified scaling with γ, pump powers, losses, and phase mismatch. Use it to compare configurations, not as a calibrated prediction of output power.

7) Why include attenuation α in the model?

Loss reduces the effective interaction length. Even with strong nonlinearity, high attenuation limits growth because power decays along the device. L_eff captures this impact in a compact form.