2.4 Parametric amplification and OPOs

L1 Phenomenon

three-wave mixing with gain; resonator threshold

A strong pump at ω_p amplifies a weak signal at ω_s and creates an idler at ω_i = ω_p − ω_s (optical parametric amplification). Inside a cavity the device oscillates once gain beats loss: an optical parametric oscillator (OPO), tunable by poling period, temperature or pump frequency. Giordmaine and Miller demonstrated tunable parametric oscillation in LiNbO₃ in 1965. On chip, TFLN rings oscillate at about 30 µW of pump reported, microdisks with Q above 10⁸ at 19.6 µW reported, and dispersion-engineered PPLN waveguides amplify by 118 dB/cm reported.

OPO in a ring: output, conversion and intracavity pump vs pump powereq. 2.15

P_s+i; conversion P_s+i/P|a_p|² / clampP_th▾ reported thresholds: 19.6 µW reported 30 µW reported

pump P in bus 100 µW

intrinsic Q_p1.0×10⁶

intrinsic Q_s = Q_i1.0×10⁶

κ_p,ex/κ_p0.50

κ_s,ex/κ_s0.50

g/2π1.00 MHz

loaded Q_p	5.00×10⁵
loaded Q_s = Q_i	5.00×10⁵
threshold P_th	23.3 µW
P / P_th	4.29
pump photons \|a_p\|²	3.74×10⁴
clamp κ_sκ_i/4g²	3.74×10⁴
signal + idler P_s+i	25.0 µW
state	oscillating
conversion P_s+i/P	25.0 %
maximum, at 4 P_th	25.0 %

H/ℏ = g(a_p^†a_sa_i + h.c.), energy decay rates, zero detunings; pump 775 nm, signal = idler 1550 nm. Defaults template; P_th, output, clamp derived. Ticks are measured devices, not computed from these inputs.

Pump power, P / P_th 4.29

Intrinsic Q, all three modes1.0×10⁶

Coupling κ_ex/κ, pump and signal

Pump power P: 100 µW
Threshold P_th: 23.3 µW
Loaded Q, pump: 5.00×10⁵
Loaded Q, signal = idler: 5.00×10⁵
g/2π: 1.00 MHz
Intracavity pump photons: 3.74×10⁴
Clamp κ_sκ_i/4g²: 3.74×10⁴
Signal + idler out: 25.0 µW
Conversion: 25.0 %
Conversion at 4 P_th: 25.0 %

Drag the pump past 1: the ring lights at P = P_th, the intracavity pump stops rising, and conversion peaks at 4 P_th. Raising Q at fixed pump power lowers P_th. Arrow lengths on one axis: their ratio is the conversion.

Q, coupling, g/2π (default 1 MHz) with H/ℏ = g(a_p^†a_sa_i + h.c.) template P_th, output, clamp: eq. 2.15 derived measured devices, not computed from these inputs: about 30 µW reported, 19.6 µW reported pump 775 nm, signal = idler 1550 nm, zero detunings; pump in spectral color, 1550 nm in false color, dashed

L2 Quantitative model

coupled modes of 2.3 L3; resonator CMT

Phase-sensitive gain

Degenerate, Δk = 0: from the coupled equations of 2.3 L3 with a fixed pump of peak power P_p, the in-phase quadrature grows as exp(Γz) and the other decays as exp(−Γz), with the normalized efficiency η_norm of the matching SHG process:

Γ = \sqrt{η_{norm} P_{p}}, \frac{G_{dB}}{L} = 20 \log_{10} (e) Γ = 8.686 \sqrt{η_{norm} P_{p}}

(2.14)

η_norm = 2600 %/W/cm² reported gives 14.0, 44.3 and 140 dB/cm at P_p = 0.1, 1 and 10 W peak derived. Non-degenerate with mismatch, no input idler (q replaces the g of the literature form, to keep g for the coupling rate):

G = 1 + \frac{Γ^{2}}{q^{2}} \sinh^{2} (q L), q^{2} = Γ^{2} - {(Δ k / 2)}^{2}

(2.14b)

At Δk = 0, G = cosh²(ΓL): 1 + (ΓL)² for small ΓL, and 6.02 dB below the phase-sensitive gain for large ΓL. Reported: > 50 dB/cm phase-sensitive in a 6 mm waveguide and > 100 dB/cm over 600 nm near 2 µm reported; 71 dB (118 dB/cm) unsaturated over 1700 to 2700 nm with 4 pJ pump pulses, 88 dB (> 146 dB/cm) saturated reported.

Parametric gain vs length

phase-sensitive, eq. 2.14cosh²(ΓL)eq. 2.14b at Δk1 + (ΓL)²

η_norm 2600 %/W/cm²

pump P_p, peak1.00 W

length L 6.0 mm

mismatch Δk0.00 rad/mm

Γ = (η_normP_p)^1/2	0.510 /mm
gain, eq. 2.14	44.3 dB/cm
G phase-sensitive	26.6 dB
de-amplified quadrature	−26.6 dB
G cosh²(ΓL), Δk = 0	20.6 dB
G at Δk	20.6 dB
Δk / 2Γ	0.000
q² = Γ² − (Δk/2)²	> 0

η_norm of measured SHG waveguides: 2600 %/W/cm² (default) reported, 4600 reported, 9600 reported; P_p, L, Δk and the shared Γ template; curves derived. Undepleted pump, no loss.

Pump peak power1.00 W

Chip length 6.0 mm

Phase mismatch Δk0.00 rad/mm

η_norm, measured SHG waveguides (%/W/cm²)

Gain axis

Γ = (η_normP_p)^1/2: 0.510 /mm
Gain per length, eq. 2.14: 44.3 dB/cm
Amplified quadrature: 26.6 dB
De-amplified quadrature: −26.6 dB
Phase-insensitive, Δk = 0: 20.6 dB
Phase-insensitive at Δk: 20.6 dB
Δk / 2Γ: 0.000

Chip lit up to the set length. Undepleted pump, no loss. Dashed: phase-sensitive gain; solid: cosh²(ΓL); dashed gray: at the set Δk; dotted: 1 + (ΓL)².

η_norm: 2600 reported, 4600 reported, 9600 reported pump power, length, Δk, shared Γ for the non-degenerate curves template gain: eq. 2.14 and 2.14b derived pump 775 nm spectral; signal 1550 nm false color

Resonator threshold

Coupled-mode theory, H/ℏ = g(a_p^†a_sa_i + h.c.), energy decay rates κ_p, κ_s, κ_i, pump input coupling κ_p,ex, all detunings zero (OPO threshold):

{| a_{p} |}_{th}^{2} = \frac{κ_{s} κ_{i}}{4 g^{2}}, P_{th} = \frac{ℏ ω_{p} κ_{p}^{2} κ_{s} κ_{i}}{16 g^{2} κ_{p, ex}}

(2.15)

Above threshold the intracavity pump clamps at that value. For κ_s = κ_i and ω_s = ω_i = ω_p/2 the summed signal and idler output is

P_{s + i} = P_{th} \frac{4 κ_{s, ex} κ_{p, ex}}{κ_{s} κ_{p}} (\sqrt{P / P_{th}} - 1)

(2.15b)

Conversion P_s+i/P peaks at P = 4 P_th with the value κ_s,exκ_p,ex/(κ_sκ_p): 25% at critical coupling, 100% only for fully overcoupled modes derived. Threshold scales as Q⁻³ when all three modes are resonant (triply resonant), and as Q⁻² for signal and idler resonant with a single-pass pump (doubly resonant, DRO) derived. A singly resonant OPO (SRO) resonates one wave only: higher threshold, but signal and idler need not both sit on resonances reported.

OPO threshold vs intrinsic Q, other inputs from the L1 OPO figureeq. 2.15

Q_p = Q_s = Q_i = QQ_s = Q_i = Q, Q_p fixedQ_p = Q, Q_s fixedticks on the P_th axis: reported 19.6 and 30 µW

all three Q (triply resonant)	−3.000
Q_s = Q_i only (DRO scaling)	−2.000
Q_p only	−1.000
P_th at the marker	23.3 µW

Slopes d log P_th / d log Q derived; marker: current inputs (L1 controls) template.

Reported devices

Device	Threshold or output	Source
TFLN PPLN ring, telecom	about 30 µW; 11% pump to signal + idler at 93 µW	reported
TFLN circuit, SHG + OPO	tens of µW; > 1 THz tuning by changing the pump frequency by hundreds of MHz	reported
microdisk, Q > 10⁸	19.6 µW	reported
dispersion-engineered PPLN, 100 ns pulses near 1 µm	tuning 1.53 to 3.25 µm, tens of mW	reported
femtosecond OPO combs	1.5 to 3.3 µm, femtojoule thresholds, up-converted to 620 nm	reported
SRO, TFLN on sapphire	1.5 to 3.3 µm; > 25 mW at 3.2 µm, 15%	reported
electrically tuned mid-IR OPO	22 THz, 2.7 to 3.4 µm	reported
frequency-modulated OPO comb	> 93% internal pump to comb, about 200 modes	reported

Tuning

Bulk PPLN OPOs (Myers et al. 1995) tune with the QPM period and the crystal temperature. Bulk CLN, pump 1064 nm: Λ = 29.91 µm phase-matches signal 1550 nm and idler 3393 nm derived, n_e from Zelmon 1997, 21 °C. Temperature tuning uses Jundt 1997, 20-250 °C. Each point solves Δk = 0 with 1/λ_p = 1/λ_s + 1/λ_i; at 21 °C the two models differ by 14 nm in signal derived.

OPO tuning, bulk CLN, pump 1064 nm, e + e → e, first orderΔk = 0, energy conserved per point

left: Zelmon 21 °C; right: signalleft: Jundt at T; right: idlerleft: Jundt at 20, 66, 112, 158, 204, 250 °C◆ Zelmon ◇ Jundt at the cursor

period Λ29.91 µm

temperature T100.0 °C

signal, Zelmon 21 °C	1549.8 nm
idler, Zelmon 21 °C	3394.2 nm
signal, Jundt at T	1560.5 nm
idler, Jundt at T	3344.2 nm
dλ_s/dT	+0.380 nm/K
dλ_i/dT	−1.744 nm/K
degenerate Λ, Zelmon	31.627 µm
degenerate Λ, Jundt at T	31.450 µm
\|1 − λ_p/λ_s − λ_p/λ_i\| here	1.1×10⁻¹⁶
maximum over both curves	1.7×10⁻¹⁶
pump ν_p	281.8 THz
signal ν_s, Jundt at T	192.1 THz
idler ν_i, Jundt at T	89.6 THz

Curves end where the idler leaves the 5 µm model window or at degeneracy (λ_s = λ_i = 2128 nm). Period thermal expansion not included. Defaults template; tuning derived.

Crystal temperature100.0 °C

Poling period29.91 µm

Signal, Jundt at T: 1560.5 nm
Idler, Jundt at T: 3344.2 nm
Signal, Zelmon 21 °C: 1549.8 nm
Idler, Zelmon 21 °C: 3394.2 nm
Degenerate Λ, Zelmon: 31.627 µm
Degenerate Λ, Jundt at T: 31.450 µm
Signal shift per kelvin: +0.380 nm/K
Idler shift per kelvin: −1.744 nm/K
|1 − λ_p/λ_s − λ_p/λ_i|: 1.1×10⁻¹⁶

Drag the temperature: the bold curve swings through the fan (20 to 250 °C, faint), signal and idler move in opposite directions, and the bar below keeps ν_s + ν_i = ν_p. Dotted: Zelmon 21 °C. Curves end where the idler leaves the 5 µm model window or at degeneracy (λ_s = λ_i = 2128 nm). Period thermal expansion not included.

n_e(λ, T): Jundt 1997 S 29.91 µm: 1550 / 3393 nm at 21 °C, Zelmon 1997 P-db derived temperature, period defaults template all waves infrared: false color, hatched and dashed

L3 Conventions and error sources

coupling-rate factors, quoted powers, resonance shifts

Hamiltonian	Threshold intracavity pump	Relative to eq. 2.15 at equal numerical g
g(a_p^†a_sa_i + h.c.), non-degenerate	κ_sκ_i/4g²	eq. 2.15
(g/2)(a_p^†a_s² + h.c.), degenerate	κ_s²/4g²	same form, κ_i → κ_s derived
g(a_p^†a_s² + h.c.), degenerate	κ_s²/16g²	4× lower derived

Eq. 2.15 uses energy decay rates and a non-degenerate Hamiltonian. Published g values are compared only after conversion to one Hamiltonian.

Threshold is quoted as on-chip pump power in the bus waveguide; fiber-coupled thresholds include facet loss.

In a doubly or triply resonant OPO, signal and idler must both sit on resonances. With unequal free spectral ranges the oscillation hops between clusters of mode pairs as the pump tunes; this is one way a pump change of hundreds of MHz can move the emission by > 1 THz reported (mechanism as interpreted here).

Photorefraction shifts resonances with power (blue shift; chapter 04) and competes with the thermal red shift during threshold sweeps; cladding removal mitigates it in TFLN rings reported.

Review: integrated PPLN OPOs, CW vs femtosecond pumping, sub-µm poling for mirrorless OPOs (Wang et al. 2026).

References

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