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Electro-Optic and Acousto-Optic Modulators

P4 specialist extension Drive a LiNbO₃ Pockels phase modulator and watch Bessel sidebands appear, bias a Mach–Zehnder modulator with complex arm fields (push-pull, chirp, extinction, eye diagram), switch a Pockels cell between polarizers, shift light with an acousto-optic Bragg cell, and steer power in an electro-optic directional coupler.

Prerequisites: Jones calculus and retarders · Mach–Zehnder interference · guided modes and coupled guides · grating equation · Bessel functions Jn

Choose a device, then use the sliders or type values into the number boxes • Every preset states what you should observe
V — —
β — —
P — —
Σ — —

Optical spectrum after the phase modulator power-normalised

Lines at ν₀ + n fm with power Jn²(β), β = πVm/Vπ. Numbers above the lines are the order n. The dashed bracket is Carson's bandwidth 2(β + 1)fm. The powers sum to 1: phase modulation only moves power between lines.

Sideband power vs modulation index

Jn²(β) for n = 0…3. The white cursor is the current β; the first carrier null is at β = 2.405 (first zero of J₀).

Field phasor eiφ(t) animated

The envelope phasor swings along the unit circle between ±β (grey arc) at fm; its length never changes, so the detected power is constant. Right: φ(t) over two drive periods.

ne (Sellmeier)
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Vπ = λ₀d/(n³rLΓ)
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β = πVm/Vπ
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Carrier J₀²
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First sidebands J₁² (each)
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Σ Jn² (all lines shown)
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Line spacing
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Peak frequency deviation βfm
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Carson bandwidth
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Vm for carrier null
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💡 How to use

Learn with this tool

P4 specialist extension This page assumes the core optics path: polarization, interference and guided waves. It models idealised devices with declared geometry and cited material constants; it is not a device-design simulator.

Learning objectives

  • Compute the half-wave voltage of a Pockels modulator from n, r, geometry and overlap, and predict the Bessel sideband powers Jn²(β) of a sinusoidally driven phase modulator.
  • Explain the Mach–Zehnder transfer curve with complex arm fields: bias point, small-signal slope π/(2Vπ), harmonic distortion, extinction ratio, chirp from unbalanced drive, and power conservation between the two output ports.
  • Use the Bragg condition, the Klein–Cook Q and coupled-mode theory to predict AOM deflection, frequency shift and efficiency, and directional-coupler transfer and switching.

Prerequisites

Model

Fields are E(t) = Re{a(t) e−iω₀t} with a slowly varying envelope a(t), so an envelope term e−inΩt sits at ω₀ + nΩ. The linear electro-optic (Pockels) effect changes an index by Δn = −½n³rE; for z-cut LiNbO₃ with TM light the relevant coefficient is r₃₃. A transverse electrode of gap d and length L with overlap Γ gives the phase Δφ = πV/Vπ. All devices are lossless, single-mode and quasi-static: the drive changes slowly compared with the optical transit time (no travelling-wave velocity mismatch).

Phase modulator: Vπ = λ₀d/(ne³r₃₃LΓ), a(t) = eiβ sin Ωt, β = πVm/Vπ, line powers Jn²(β), Σ Jn² = 1

MZM: Pout/Pin = ½[1 + s cos(πV/Vπ + φbias)], s = 2√(r(1−r)) → cos²(πV/2Vπ + φbias/2) for r = ½

Chirp: α = [(s₁ − s₂)/(s₁ + s₂)] cot(Δφ/2); push-pull α = 0

Pockels cell: T⊥ = sin²(Γ/2), Γ = Γ₀ + πV/Vπ; KDP longitudinal Vπ = λ₀/(2no³r₆₃)

AOM: Λ = v/fa, θB ≈ λ₀/(2Λ), νm = ν₀ + m fa, Q = 2πλ₀L/(nΛ²), η₁ = sin²(ν/2) with ν = (√2π/λ₀)√(M₂PaL/H)

Coupler: P₂(z) = (κ²/g²) sin²(gz), g = √(κ² + (Δβ/2)²), Lc = π/(2κ)

λ₀, ν₀
vacuum wavelength and optical frequency of the carrier
ne, no
extraordinary and ordinary indices (LiNbO₃ from the Zelmon 1997 Sellmeier; KDP no = 1.51)
r₃₃, r₁₃, r₆₃
Pockels coefficients: LiNbO₃ 30.8 and 8.6 pm/V, KDP 10.5 pm/V (unclamped, dispersion neglected)
d, L, Γ
electrode gap, electrode length, RF–optical overlap factor
Vπ
half-wave voltage: π phase (phase modulator), full on–off swing (MZM, Pockels cell)
β, fm
phase-modulation index and drive frequency
s₁, s₂, r
arm drive weights (arm 1 gets +s₁V, arm 2 gets −s₂V) and input-splitter power ratio
α
small-signal chirp (Henry) parameter dΦ/dt ÷ d ln|a|/dt, engineering sign convention
Λ, v, fa, M₂
acoustic wavelength, sound speed, frequency, acousto-optic figure of merit (s³/kg)
Q, ν
Klein–Cook thickness parameter and Raman–Nath phase-modulation depth
κ, Δβ
coupling coefficient and propagation-constant mismatch β₁ − β₂ of the two guides
Derivation: Bessel sidebands and the MZM transfer curve

The generating function of the Bessel functions is eiβ sin θ = Σk Jk(β)eikθ. With θ = Ωt and the carrier e−iω₀t, the term k = −n is at ω₀ + nΩ with amplitude J−n(β) = (−1)nJn(β). Parseval's theorem on one drive period gives Σ|Jn|² = ⟨|eiφ|²⟩ = 1. The tool checks the Bessel amplitudes against an FFT of the sampled envelope.

In the MZM the input splitter sends √r and i√(1−r) into the arms; the arms add φ₁ = s₁πV/Vπ,PM + φbias/2 and φ₂ = −s₂πV/Vπ,PM − φbias/2; the output 50/50 coupler (1/√2)[[1, i], [i, 1]] gives aout = i(√r eiφ₁ + √(1−r) eiφ₂)/√2. Then |aout|² = ½[1 + 2√(r(1−r)) cos(φ₁ − φ₂)], and the other port carries the rest, because the coupler is unitary. For r = ½ and Δφ = πV/Vπ + φbias this is cos²(πV/2Vπ + φbias/2). Its slope is largest at quadrature (Δφ = ±π/2) where |dP/dV| = π/(2Vπ). For the sine drive at quadrature P = ½ ∓ ½ sin(β′ sin Ωt), β′ = πVm/Vπ, so the detected harmonics are J₁(β′), J₃(β′), … and HD3 ≈ β′²/24 for small drive: the curve is odd about quadrature, so even harmonics vanish.

Chirp: for r = ½, aout = i ei(φ₁+φ₂)/2 cos(Δφ/2). The phase (φ₁ + φ₂)/2 moves only if s₁ ≠ s₂, and dividing its time derivative by d ln|a|/dt = −½ tan(Δφ/2) dΔφ/dt gives α = [(s₁ − s₂)/(s₁ + s₂)] cot(Δφ/2) in the engineering (e+iωt) sign convention; sign conventions differ between textbooks, but |α| and its sign flip between the two quadrature points do not.

Derivation: Bragg diffraction, Klein–Cook Q and coupled modes

A sound wave n(x, t) = n + Δn cos(Kx − Ωt) is a moving phase grating. Order m has transverse wavevector kx + mK and, from energy conservation with the moving grating (Doppler shift), frequency ω₀ + mΩ. Writing the field as Σ Bm(z) ei(kx+mK)x and keeping the paraxial phase mismatch of each order gives the Raman–Nath equations dBm/dζ = (ν/2)(Bm−1 − Bm+1) − i(Q/2)m(m − a)Bm, ζ = z/L, with ν = k₀ΔnL and a = θin/θB. Q = K²L/k = 2πλ₀L/(nΛ²) measures how fast the higher orders dephase. For Q → 0 the Bessel recursion 2Jm′ = Jm−1 − Jm+1 gives |Bm|² = Jm²(ν). For Q → ∞ at a = 1 only orders 0 and +1 are phase-matched, and B₁ = sin(νζ/2). The tool integrates the full system with a matrix exponential (unit-norm because the matrix is anti-Hermitian) and the tests cross-check it with RK4. The index amplitude is Δn = √(M₂Ia/2) with acoustic intensity Ia = Pa/(LH), so ν = k₀ΔnL = (√2π/λ₀)√(M₂PaL/H).

Coupler: with A₁′ = −iκA₂eiΔβz and A₂′ = −iκA₁e−iΔβz, substituting A = e±iΔβz/2u gives constant coefficients with eigenvalues ±ig, g = √(κ² + Δβ²/4). The cross power is (κ/g)² sin²(gz), so a mismatch shortens the beat length and caps the transfer at κ²/g² = 1/(1 + (Δβ/2κ)²). A coupler of length Lc reaches the bar state when gLc = π, i.e. Δβ = √3π/Lc. Reversing Δβ halfway (Kogelnik–Schmidt) makes the cross state reachable electrically for any L between Lc and 3Lc.

Exercise 1: put the carrier to zero

  1. With the default electrodes (d = 10 µm, L = 20 mm, Γ = 0.5, λ₀ = 1550 nm), predict Vπ and the drive amplitude Vm that removes the carrier.
  2. Choose Phase modulator and type your Vm into the number box.
  3. Read J₀², J₁² and Σ Jn² from the readouts and the line spacing from the spectrum.
  4. Where did the carrier power go, and why does the total stay 1?
Show answer

ne = 2.1376, so Vπ = 1.55 µm × 10 µm / (9.767 × 30.8 pm/V × 20 mm × 0.5) = 5.15 V. The carrier vanishes at β = 2.405, Vm = 2.405 Vπ/π = 3.94 V. Then J₁² = 0.269 in each first sideband, J₂² = 0.186, J₃² = 0.039 and the lines stay 10 GHz apart. Phase modulation only redistributes power among the lines: |eiφ(t)| = 1 at every instant, so Parseval forces Σ Jn² = 1.

Exercise 2: limiting cases of the acousto-optic cell (thin vs thick grating)

  1. Load AOM Raman–Nath (Q ≈ 0.04) and then AOM Bragg regime (Q ≈ 12). For each, predict how many orders carry power and what η₁ should be from the thin (J₁²) and thick (sin²(ν/2)) formulas.
  2. Compare the bars with the dots in the order plot and read η₁ from the readouts. Then set the incidence to 0 in the Bragg case.
  3. Record η₁ (coupled-wave), J₁²(ν) and sin²(ν/2) for both presets; read the frequency of order +1.
  4. Which formula applies in which limit, and why does normal incidence kill the diffraction at large Q?
Show answer

At Q ≈ 0.04 the coupled-wave bars sit on the Bessel dots (orders ±1, ±2 symmetric, η₁ = J₁²(ν)); the Bragg formula is wrong there. At Q ≈ 12 with Bragg incidence only orders 0 and +1 matter and η₁ is within a few per cent of sin²(ν/2); it converges to it as Q → ∞. At normal incidence and large Q, orders ±1 are both mismatched by Q/2, so almost nothing diffracts. Order +1 is shifted up by exactly fa in every case.

Exercise 3: linear modulation at quadrature

  1. Load MZM quadrature, linear. Predict the slope dP/dV and the third-harmonic distortion HD3 for Vm = 0.5 V.
  2. Change the bias by ±30° and double Vm.
  3. Record HD2 and HD3 for each setting.
  4. Why is HD2 zero only at quadrature, and why does HD3 grow by about 12 dB when Vm doubles?
Show answer

Push-pull Vπ = 2.58 V, slope π/(2Vπ) = 0.610 V⁻¹. β′ = π·0.5/2.58 = 0.610, HD3 = 20 log(J₃/J₁) ≈ 20 log(β′²/24) = −36.0 dBc. The cos² curve is odd about the quadrature point, so even harmonics cancel there; off quadrature the curvature adds HD2. HD3 ∝ β′², so doubling the drive adds 20 log 4 ≈ 12 dB.

Exercise 4: switching a directional coupler

  1. Load Coupler Δβ = 2κ. Predict the maximum power that ever reaches guide 2. Then predict the voltage that switches a phase-matched L = Lc coupler fully to the bar state.
  2. Read the z plot; then load EO switch to bar.
  3. Record the peak of P₂(z) and the output powers at the switching voltage.
  4. Why can the uniform coupler reach the bar state but not always the cross state, and how does Δβ-reversal fix that?
Show answer

Δβ/2κ = 1 gives a maximum transfer 1/(1 + 1) = 50 %. The bar state needs gLc = π, g = 2κ, so Δβ = 2√3κ = √3π/Lc. With Lc = 5 mm, d = 10 µm and Γ = 0.5 this is ≈ 17.8 V. The cross state needs Δβ = 0 and L an odd multiple of Lc, which fabrication rarely hits exactly; reversing Δβ halfway adds a second tuning knob, so a voltage exists that gives the cross state for L from Lc to 3Lc (try Δβ-reversal, L = 2Lc).

Worked example: a push-pull LiNbO₃ MZM for 10 Gb/s

Take z-cut LiNbO₃ at λ₀ = 1550 nm with TM light, electrode gap d = 10 µm, length L = 20 mm, overlap Γ = 0.5.

  1. Index: the Zelmon Sellmeier gives ne = 2.1376, so ne³ = 9.767. With r₃₃ = 30.8 pm/V, one arm electrode needs Vπ,PM = λ₀d/(n³rLΓ) = 5.15 V.
  2. Push-pull (s₁ = s₂ = 1) doubles the phase difference per volt: Vπ = 2.58 V. At quadrature the small-signal slope is π/(2Vπ) = 0.61 per volt, i.e. 61 % of the input power per volt.
  3. Because s₁ = s₂, the common phase (φ₁ + φ₂)/2 never moves: α = 0 (chirp-free). Driving only one arm (s₂ = 0) doubles Vπ to 5.15 V and gives |α| = 1 at quadrature.
  4. A 10 Gb/s NRZ drive with Vpp = Vπ about quadrature swings between the null and the peak. With an ideal splitter the static extinction is infinite; a real 25 dB device limits the dynamic extinction to 25 dB.
  5. If the driver has a 10–90 % rise time of 0.5 bit periods (50 ps), the single-pole filter leaves the "1" level at ≈ 0.989 and the "0" at ≈ 0.012: the vertical eye opening is ≈ 0.95 and the dynamic extinction ≈ 19 dB (preset 10 Gb/s NRZ eye).

Load 10 Gb/s NRZ eye and compare each number with the readouts.

When the model fails

  • Quasi-static drive: real high-speed modulators are travelling-wave devices. RF loss and the velocity mismatch between the microwave (index ≈ 4.2 for a plain coplanar line on LiNbO₃) and the light (group index ≈ 2.2) limit the bandwidth. Here the modulator responds instantly; only the electrical driver has a bandwidth.
  • Constant material data: the r coefficients are fixed low-frequency values without dispersion; clamped (high-frequency) values differ by a few per cent. Temperature, photorefractive damage, DC bias drift and piezo-electric ringing are not modelled.
  • Pockels cell: the transverse LiNbO₃ cell has a large natural birefringence 2πL(ne − no)/λ₀ that must be compensated; Γ₀ is the residual after compensation. Beam divergence (off-axis rays) and finite polarizer extinction are ignored.
  • AOM: plane-wave light and sound; no beam divergence (which relaxes the Bragg condition), acoustic attenuation, anisotropic diffraction or rise time (sound transit across the beam). M₂ values are for longitudinal waves with the optimum polarization.
  • Coupler: weak-coupling CMT with lossless identical guides; κ does not depend on voltage and there are no radiation losses. For strong coupling use the exact supermodes (waveguide tool).

References

  • A. Yariv and P. Yeh, Photonics, 6th ed., Oxford (2007): Ch. 9 (electro-optic modulation), Ch. 12 (acousto-optics), Ch. 13 (coupled modes, directional couplers).
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019): Ch. 20 (acousto-optics), Ch. 21 (electro-optics).
  • E. L. Wooten et al., "A review of lithium niobate modulators for fiber-optic communications systems," IEEE JSTQE 6, 69 (2000).
  • D. E. Zelmon, D. L. Small and D. Jundt, "Infrared corrected Sellmeier coefficients for congruently grown lithium niobate," JOSA B 14, 3319 (1997).
  • R. S. Weis and T. K. Gaylord, "Lithium niobate: summary of physical properties and crystal structure," Appl. Phys. A 37, 191 (1985).
  • W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
  • H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating Δβ," IEEE JQE 12, 396 (1976).
  • F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87 (1988).