Tool · Silicon modulators

Plasma Dispersion Calculator

How much do free electrons and holes change the refractive index and the absorption of silicon, and how long must a phase shifter be to reach π, at what cost in loss? The calculator evaluates the Soref and Bennett relations at 1310 and 1550 nm and the Nedeljkovic fits at 1550 and 2000 nm for any electron and hole density, added or removed, and converts Δn and Δα into phase, loss, the length for π and the loss of doped waveguides for a given length and mode overlap. Background: plasma dispersion effect, free-carrier absorption, and Mach-Zehnder modulator.

Carriers and wavelength

Wavelength

Relation

Carrier change

Electron density ΔN (cm⁻³)

Hole density ΔP (cm⁻³)

carriers added by injection or removed by depletion (set the direction above), or the dopant density of a doped waveguide; below 10¹⁴ the contribution is taken as zero

Mode overlap Γ

fraction of the mode power inside the region whose carrier density changes; 1 for a uniformly filled core, 0.1 to 0.4 for the depleted slice of a pn junction

Phase-shifter length L (µm)

Presets

The relations are empirical fits to the absorption spectra of doped silicon with a Kramers-Kronig transform for the index, so they apply to dopants as well as to injected or depleted carriers. Densities are spatially uniform in the modulated region; a real junction has a profile, which the overlap factor stands in for.

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Readouts

Dopant density	n-type loss	p-type loss

Index change versus carrier density

electronsholes

Absorption change versus carrier density

electronsholes

Loss of a π phase shifter versus carrier density

electrons onlyholes onlyequal electrons and holes

Checked against

These checks run in your browser on every load. The relations are evaluated at the densities the sources quote and compared with the published coefficients times the density, the two 1550 nm relations are compared with each other, and the unit conversions are checked against hand-computed values.

Check	Expected	Computed	Tolerance

The table evaluates each empirical relation at the density its source quotes and compares it with the published coefficients of Soref and Bennett (1987, as quoted by Reed et al., 2010) and Nedeljkovic et al. (2011), then checks the unit conversions and the depletion arithmetic against hand-computed values. Tolerances are relative to the expected value; where a row lists several values, the tolerances follow in the same order, and “condition” marks the one row whose Expected entry is itself the test.

The model

Free carriers lower the refractive index of silicon and add absorption. Both effects are taken from empirical fits of the form

Δ n = - (a N^{b} + c P^{d})

Δ α = e N^{f} + g P^{h}

with the electron and hole densities $N$ and $P$ in $c m^{- 3}$ and $Δ α$ in $c m^{- 1}$ . At 1550 nm the Soref and Bennett set is

Δ n = - 8.8 \times 10^{- 22} N - 8.5 \times 10^{- 18} P^{0.8}

Δ α = 8.5 \times 10^{- 18} N + 6.0 \times 10^{- 18} P

and the Nedeljkovic sets use the same form with fitted exponents. Removing carriers (depletion) reverses the sign of both changes. A mode that overlaps the carriers by a fraction $Γ$ over a length $L$ acquires

φ = 2 π Γ Δ n L / λ

L_{π} = \frac{λ}{2 Γ ∣ Δ n ∣}

l o s s = 4.343 Γ Δ α L d B

The relations are fits to measured absorption spectra of crystalline silicon at room temperature, with the index change obtained through the Kramers–Kronig relation, so they carry the uncertainty of that data and are extrapolations outside the fitted density range, about $10^{17}$ to $10^{20} c m^{- 3}$ ; the readout panel says so when a nonzero density falls below it. A single $Γ$ treats the carrier distribution as uniform over the overlapping part of the mode; a real PN or PIN junction has a nonuniform, bias dependent profile that a device simulation resolves. Heating from injection current, which raises the index through the thermo-optic effect, and the speed of the carrier response are outside the model.

Worked example

The default preset injects $10^{18} c m^{- 3}$ electrons and holes at 1550 nm with $Γ = 1$ and $L = 500 μ m$ . Electrons give $Δ n = - 8.80 \times 10^{- 4}$ and holes $- 2.135 \times 10^{- 3}$ (since $10^{18 \times 0.8} = 10^{14.4}$ ), a total of $- 3.015 \times 10^{- 3}$ of which 71 % comes from holes. The absorption rises by 8.5 + 6.0 = 14.5 $c m^{- 1}$ , or 63.0 dB/cm. Then $L_{π} = λ / (2 ∣ Δ n ∣) = 257 μ m$ and the loss over that length is 1.62 dB. Over the full 500 µm the phase is 6.11 rad (1.95π) and the loss 3.15 dB, the values the readout panel shows.

Related: the Vπ fitter extracts Vπ from a measured modulator transfer curve, and the article on measuring modulator Vπ and electro-optic bandwidth describes the bench procedure.

References: R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE Journal of Quantum Electronics 23, 123–129 (1987). M. Nedeljkovic, R. Soref and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1–14 µm infrared wavelength range,” IEEE Photonics Journal 3, 1171–1180 (2011). G. T. Reed et al., “Silicon optical modulators,” Nature Photonics 4, 518–526 (2010).