2.3 Second-order nonlinear optics

L1 Phenomenon

SHG 1550 → 775 nm, bulk CLN and TFLN

The polarization of LN contains a term quadratic in the optical field (χ⁽²⁾), so two photons merge into one at the sum frequency (SFG; SHG when both are equal) or one photon splits into two (DFG; parametric amplification). The largest coefficient, d₃₃, couples fields all polarized along c.

Dispersion makes 775 nm light slower than 1550 nm light in bulk CLN: the second harmonic slips out of phase after the coherence length L_c = π/Δk = 9.495 µm derived and converts back. Reversing the ferroelectric domain every L_c (period Λ = 18.99 µm, PPLN) restarts the growth in every domain, and the harmonic power grows as L². In a TFLN waveguide etched from a sub-micrometre film the period is near 4 µm, set by waveguide dispersion reported; measured normalized efficiencies reach thousands of percent per watt per square centimetre (L2).

SHG along z, 1550 → 775 nm, e + e → eRK4 · eq. 2.13 · tanh²

d(z) |±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±|
     0                                              z = L = 10.0 mm

RK4, coupled-mode equationsundepleted, eq. 2.13P₀ tanh², QPM

Poling

Unpoled Δk

η_norm, %/W/cm²

Pump P_ω(0)10.0 mW

Length L 10.0 mm

Duty cycle D0.50

Loss, both waves0.00 dB/cm

P axis

P_2ω(L), RK4	2.208e−03 W	derived
P_2ω(L) / P_ω(0)	2.208e+01 %	derived
eq. 2.13, undepleted	2.600e−03 W	derived
P_ω(L) / P_ω(0)	77.922 %	derived
η_norm (input)	2600 %/W/cm²	reported
η_norm sin²(πD) L², QPM	2600.0 %/W	derived
L_c = Λ/2 (unpoled)	9.495 µm	derived
\|P_ω + P_2ω − P₀\| / P₀, lossless	9.5e−15	derived

η_norm: measured values (L2 table) reported; loss and D are inputs template. QPM: domain-averaged coupling. Unpoled: full d₃₃ with the bulk or TFLN mismatch; the harmonic oscillates with period 2L_c. Strip: 1 character = L/60; ± = domains finer than one character.

from z = 0 at 2 px per µmΛ = 18.989 µm, D = 0.50

50 µm

Drawntop: the guide from z = 0 at 2 px per µm (at most the first 450 µm); middle: the whole guide, 775 nm glow scaled to its maximum in each view; dashed: 1550 nm pump, width = remaining power. Poling off: the glow pulses every coherence length. Periodic poling: it grows monotonically; at watts the curve bends from L² to tanh² as the pump thins. Curves: RK4 of the coupled-mode equations (solid), undepleted eq. 2.13 (dashed), P₀ tanh² (dotted, QPM). QPM uses the domain-averaged coupling κ² = η_norm sin²(πD); without poling the full d₃₃ acts with the bulk or TFLN mismatch. Loss and duty cycle are inputs template.

η_norm measured reported curves: coupled-mode RK4, eq. 2.13, tanh² derived pump 1550 nm: false color, dashed; SH 775 nm: deep red

Poling

Mismatch without poling

Measured η_norm, %/W/cm²

Power axis

Pump power10.0 mW

Length 10.0 mm

Duty cycle0.50

Loss, both waves0.00 dB/cm

Second harmonic at L, RK4: 2.208e−03 W
Converted P_2ω/P_ω(0): 2.208e+01 %
Undepleted, eq. 2.13: 2.600e−03 W
Pump remaining: 77.922 %
Measured η_norm (input): 2600 %/W/cm² reported
Low-power efficiency, QPM: 2600.0 %/W
Coherence length L_c (unpoled): 9.495 µm
Energy check, lossless: 9.5e−15

L2 Quantitative model

3m d tensor, eq. 2.10 to 2.13

d tensor

Contracted index l: 1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy. Point group 3m with Kleinman symmetry (d₁₅ = d₂₄ = d₃₁; Kleinman 1962):

d = (\begin{matrix} 0 & 0 & 0 & 0 & d_{31} & - d_{22} \\ - d_{22} & d_{22} & 0 & d_{31} & 0 & 0 \\ d_{31} & d_{31} & d_{33} & 0 & 0 & 0 \end{matrix})

(2.10)

P_{i} (2 ω) = 2 ε_{0} \sum_{l} d_{i l} {(E_{x}^{2}, E_{y}^{2}, E_{z}^{2}, 2 E_{y} E_{z}, 2 E_{x} E_{z}, 2 E_{x} E_{y})}_{l}

(2.11)

Complex-amplitude convention of Boyd 2008; χ⁽²⁾ = 2d. For two distinct input frequencies (SFG) the prefactor is 4ε₀.

Values (pm/V)

No source held tabulates a post-1997 absolute d set: every value below is a legacy-scale compilation, a secondary table without per-cell references, or unverified. No set is used as default.

Source	Fundamental	d₃₃	d₃₁	d₂₂	Scale	Status
Volk & Wohlecke Table 1.2 (CLN), citing Nikogosyan 2005	1.06 µm	34	6	3	legacy	P-comp
same, near-SLN	1.06 µm	42	5	2.5	legacy	P-comp
Zhu et al. review Table 1 (no per-cell reference)	1064 nm	−27.0	−4.3	2.10	not stated	S
Shoji et al. 1997, absolute scale	1.064 µm	25.2 claimed	about 4.6 claimed	–	post-1997	U
Kitamoto et al. 1995, parametric fluorescence	488 / 532 nm pump	–	4.8 ± 0.5 / 4.3 ± 0.5 claimed	–	post-1997 method	U

Weis & Gaylord 1985 has no SHG d table. The legacy d₃₃ = 34 pm/V is 35% above the claimed 25.2 pm/V U, which attributes the older values to multiple-reflection errors (claim, abstract only). Miller's rule scales χ⁽²⁾ with the product of the linear susceptibilities at the three frequencies; Shoji et al. report that it holds poorly for LN (claim) U.

Field mapping (eq. 2.10, no further assumption)

Cut, mode	Fields	Active coefficient	Phase matching
x-cut, TE, propagation y	E along z at ω and 2ω	d₃₃	QPM, type 0 (e + e → e)
z-cut, TM	E mostly along z	d₃₃	QPM, type 0
z-cut, TE, in-plane field at α from x	E in xy	TE SH: −2ε₀d₂₂E² sin 3α, largest for E along y (propagation x), zero for E along x (propagation y); TM SH via d₃₁	modal or QPM
bulk, type I (o + o → e)	ordinary inputs, no z component	d₃₁, d₂₂ only	birefringent, angle-tuned

Birefringent phase matching never uses d₃₃: ordinary inputs have no z component, so only the xx, yy, xy columns act.

d tensor: which coefficient a cut and polarization usessymbolic

Cut

Input field

Propagation in the plane 0°

Coefficient set

P/(2ε₀E²)	× d₃₃	pm/V
x	0.000	–
y	0.000	–
z	+1.000	–
onto TE out	+1.000	–
onto TM out	0.000	–

	1 xx	2 yy	3 zz	4 yz	5 xz	6 xy
x	0	0	0	0	d31	−d22
y	−d22	d22	0	d31	0	0
z	d31	d31	d33	0	0	0

            z               
            |               
            |               o: E along x (TM)
            |               +: origin
············+----------ky   k at   0° from y
            |               
            |               
            |               
x-cut, normal x

k ( 0.00, +1.00, 0.00)E ( 0.00, 0.00, +1.00)active: d33

Propagation angle from the first in-plane axis (x-cut: y; y-cut and z-cut: x) toward the second. TE: in-plane field perpendicular to k; TM: field along the wafer normal. z-cut TE puts E in the xy plane, the field of an ordinary wave in bulk. Inverted cells: coefficients the input drives. Rotating z-cut TE through 180° never drives d₃₃, and the TE response repeats every 120°. Numbers: 1.06 µm values of the selected set.

z-cut TE puts E in the xy plane, the field of an ordinary wave in bulk: rotating it through 180° never drives d₃₃, only d₃₁ and d₂₂, and the TE response repeats every 120°. x-cut TE is driven by d₃₃ alone. Propagation angle from the first in-plane axis (x-cut: y; y-cut and z-cut: x) toward the second; TE: in-plane field perpendicular to k; TM: field along the wafer normal. Numbers: 1.06 µm values of the selected set.

input 1550 nm: false color, dashed, double arrow = oscillating field radiated P(2ω), 775 nm: deep red, drawn downstream along k; with symbols, one arrow per coefficient

Crystal cut

Input field

Propagation in the plane 0°

Coefficients symbolic

Propagation k: ( 0.00, +1.00, 0.00)
Field E: ( 0.00, 0.00, +1.00)
P/(2ε₀E²): d33 ( 0.00, 0.00, +1.00)
SH field along TE: d33
SH field along TM: 0
Coefficients driven: d33

Phase matching

Wave-vector mismatch with an m-th order grating of period Λ (phase matching at Δk = 0):

Δ k = k_{3} - k_{1} - k_{2} - \frac{2 π m}{Λ}

(2.12)

Birefringent, type I SHG, bulk CLN, 21 °C. 1550 nm: θ_pm = 49.55° from c, SH walk-off 2.034° derived; 2000 nm: 45.01° derived. 1064 nm: no angle exists, since n_e(532 nm) − n_o(1064 nm) = +1.46×10⁻³ > 0 derived; 1064 nm type-I SHG in CLN needs temperature tuning (noncritical phase matching).

Quasi-phase matching. Order m, duty cycle D (Fourier coefficient of the square-wave sign of d; form as in Fejer et al. 1992), so (2/π) d₃₃ at D = 0.5, m = 1:

d_{eff} = \frac{2}{m π} d_{33} \sin (m π D)

Process, bulk CLN, e + e → e, 21 °C	Λ (µm)	L_c = Λ/2 (µm)	Status
SHG 1064 → 532 nm	6.818	3.409	derived
SHG 1550 → 775 nm (also degenerate OPO pumped at 775 nm)	18.989	9.495	derived
SHG 2090 → 1045 nm	31.07	15.54	derived
DFG / OPO: pump 1064, signal 1550, idler 3393.4 nm	29.91	14.96	derived

Jundt n_e at 24.5 °C gives 18.976 µm for 1550 nm SHG derived. TFLN waveguides shift the period to a few µm because n_eff disperses differently from bulk n_e; x-cut films have been poled at 600 nm reported and 747 nm periods reported. Early thin-film PPLN converters: Chang et al. 2016.

Undepleted and depleted SHG

P_{2 ω} = η_{norm} P_{ω}^{2} L^{2} {sinc}^{2} (\frac{Δ k L}{2}), η_{norm} = \frac{8 π^{2} d_{eff}^{2}}{ε_{0} c n_{ω}^{2} n_{2 ω} λ_{ω}^{2} A_{eff}}

(2.13)

η_norm computed from d₃₃ inherits the d spread. Eq. 2.13 at 1550 nm with bulk indices, A_eff = 1 µm² template, d_eff = (2/π) d₃₃:

d₃₃ source	d₃₃ (pm/V)	d_eff (pm/V)	η_norm (%/W/cm²)	Status
Volk & Wohlecke, CLN, legacy scale P-comp	34.0	21.6	5828	derived
same, near-SLN P-comp	42.0	26.7	8893	derived
Zhu et al. review, \|d₃₃\| S	27.0	17.2	3675	derived
Shoji et al. 1997, claimed U	25.2 claimed	16.0	3201 claimed	U

Depleted, at Δk = 0 (Armstrong et al. 1962):

P_{2 ω} = P_{0} \tanh^{2} (\sqrt{η_{norm} P_{0}} L)

Bulk focusing cannot reach the L² law: a Gaussian beam diffracts, the optimum confocal parameter is b = L/2.84 reported, and P_2ω then grows only linearly in L. Waveguides hold A_eff fixed over centimetres, hence the L² law and the 1/A_eff gain.

Measured normalized efficiencies

Sort

Device	Value	Normalization	Date	Status
LiNbO₃ waveguide, field-poled at room temperature, first-order QPM, blue SHG	600 %/W/cm²; 20.7 mW blue	waveguide	1993-02-01	reported
TFLN PPLN, 1.5 µm pump, 4 µm period	2600 %/W/cm²	waveguide	2018-11-07	reported
TFLN, shallow etch, MgO, 5 mm	3757 %/W/cm² (939 %/W)	waveguide	2020-06-19	reported
TFLN, actively monitored poling	4600 %/W/cm²	waveguide	2019-08-28	reported
double-layer, polarization-reversed TFLN	9600 %/W/cm² measured; >10,000 theory; 85% absolute at high power	waveguide	2024-11-07	reported
TFLN, blue SHG	33,000 %/W/cm²	waveguide	2022-05-19	reported
TFLN on sapphire, DFG to 3.66 µm	up to 200 %/W/cm²	waveguide (DFG)	2021-06-16	reported
TFLN, modal phase matching (no poling), 8 mm	4.7 %/W	waveguide, %/W	2018-08-16	reported
z-cut PPLN microring	250,000 %/W	ring	2019-11-21	reported
PPLN microring; g/2π about 1.2 MHz	5,000,000 %/W	ring	2020-11-23	reported

Waveguides: %/W/cm² = P_2ω/(P_ω²L²); rings: %/W. The two units are not convertible without a model (L3). Dates: issue dates.

Acceptance bandwidths

sinc²(x/2) falls to one half at x = ±2.783, so the full width at half maximum in any parameter q is

δ q = \frac{2 \times 2.783}{L | \partial Δ k / \partial q |} = \frac{5.566}{L | \partial Δ k / \partial q |}

SHG pump, bulk CLN, first-order QPM	FWHM in λ_ω × L, Zelmon 21 °C (nm cm)	FWHM in T × L, Jundt 25 °C (K cm)	dλ_ω/dT, Jundt 25 °C (nm/K)
1064 nm	0.205 derived	3.21 derived	0.064 derived
1550 nm	1.176 derived	10.12 derived	0.115 derived

Thermal expansion of the period not included. The temperature acceptance shifts with the reference temperature: 1550 nm SHG 10.14 K cm at 24.5 °C, 10.26 K cm at 21 °C derived.

Group-velocity mismatch, bulk, e-polarized 1550/775 nm: Δn_g = 0.0905, 302 fs/mm derived. Comparison with a TFLN device: Zhao et al. report a 1 nm linewidth for 5 mm and 0.1 nm/°C tuning reported; bulk CLN scaled to 5 mm gives 2.35 nm derived and 0.115 nm/K derived. Waveguide dispersion lets nanophotonic PPLN cancel group-velocity mismatch and run femtosecond SHG with < 100 fJ reported.

Phase matching, bulk CLN, eq. 2.12e + e → e QPM; o + o → e BPM

Pump (SHG)

Process

λ₁1.550 µm

λ₂ (SFG, DFG)1.064 µm

QPM order m

Duty cycle D0.50

Length L 10.0 mm

Operating T 40.0 °C

Period offset ΔΛ 0.0 nm

n_e for Λ

λ_SH	0.775 µm	derived
Λ, order m = 1	18.989 µm	derived
L_c = π/Δk	9.495 µm	derived
d_eff/d₃₃ = 2 sin(mπD)/(mπ)	0.6366	derived
θ_pm type I (o + o → e)	49.55°	derived
walk-off ρ of the e wave	2.034°	derived
FWHM in λ_ω at L (SHG)	1.176 nm	derived
FWHM in T at L (SHG, Jundt)	9.64 K	derived
dλ_ω/dT at fixed Λ (SHG, Jundt)	0.1205 nm/K	derived
GVM 1/v_g,2ω − 1/v_g,ω (SHG)	301.9 fs/mm	derived
grating Λ_g = mΛ(40 °C) + ΔΛ (SHG, Jundt)	18.9264 µm	derived
Δk at λ_ω, T, Λ_g (eq. 2.12)	0.000 rad/mm	derived
ΔkL/2	0.000 rad	derived
sinc²(ΔkL/2)	1.0000	derived
SH power / matched 1st order, D = 0.5	1.0000	derived
FWHM in Λ at L	31.73 nm	derived

Curve versus

bulk CLN, order 1, Λ 18.989 µm, D 0.50
|+++++++++----------+++++++++----------+++++++++---|
TFLN, Λ 4 µm reported (Wang et al. 2018, 1.5 µm pump)
|++--++--++--++--++--++--++--++--++--++--++--++--++|
 0         10        20        30        40        50 µm
SH 775 nm |########################################| 1.000

λ₁: fundamental (SHG), first input (SFG), pump (DFG); λ₂: second input (SFG), signal (DFG, idler from energy conservation). Fixed grating Λ_g = mΛ₁(λ_ω, 40 °C, Jundt) + ΔΛ, operated at T: the curve shows sinc²(ΔkL/2) of that grating versus the selected parameter, band = FWHM, marker = operating point; the bar shows the generated SH power relative to a matched first-order grating with D = 0.5 derived. Jundt n_e is defined from 20 to 250 °C, so a curve ends where T leaves that window. Period expansion with T not included. BPM angle: Zelmon 21 °C. TFLN strip: reported period reported.

Bulk CLN, order 1, computedΛ = 18.989 µm = 38.0 px

TFLN waveguide, reportedΛ ≈ 4 µm at 1.5 µm pump = 8 px

50 µm at 2 px per µm

Drawndomains at 2 px per µm: the computed bulk period (1550 nm SHG: 38 px) above the reported TFLN period (8 px). Below, a fixed grating Λ_g = mΛ₁(40 °C, Jundt) + ΔΛ: the generated beam, in the color of its wavelength, is as bright as sinc²(ΔkL/2) allows; heat the crystal or offset the period and it fades, then flickers through the side lobes. The curve shows the same sinc² versus the selected parameter, area inside the FWHM filled, diamond = operating point. First wavelength: fundamental (SHG), first input (SFG) or pump (DFG); second: second input (SFG) or signal (DFG, idler from energy conservation). Curve and beam: Jundt n_e, defined from 20 to 250 °C, so a curve ends where T leaves that window; brightness relative to a matched first-order grating with D = 0.5. Period expansion with T not included. Type-I angle: Zelmon 21 °C.

periods, angles, bandwidths derived temperature values: Jundt derived detuned beam and curve derived TFLN period reported IR: false color, dashed

Pump, SHG

Process

Fundamental, input 1 or pump1.550 µm

Input 2 or signal1.064 µm

QPM order

Duty cycle0.50

Crystal length 10.0 mm

Operating temperature 40.0 °C

Period offset ΔΛ 0.0 nm

Index model for the period

Curve versus

Output wavelength: 0.775 µm
Period Λ, order 1: 18.989 µm
Coherence length: 9.495 µm
d_eff / d₃₃: 0.6366
Type-I angle from c: 49.55°
FWHM in wavelength: 1.176 nm
FWHM in temperature: 9.64 K
Tuning dλ/dT: 0.1205 nm/K
Group-velocity mismatch: 301.9 fs/mm
Grating Λ_g: 18.9264 µm
Δk: 0.000 rad/mm
ΔkL/2: 0.000 rad
sinc²(ΔkL/2): 1.0000
SH power / matched 1st order: 1.0000
FWHM in Λ: 31.73 nm

L3 Normalizations and conventions

%/W/cm², %/W, absolute %, pulse energy

Why the efficiency numbers do not compare directly

Normalization	Definition	Valid when	Breaks when
%/W/cm² (η_norm)	P_2ω/(P_ω²L²), low-power slope, phase matched	uniform QPM, no loss, undepleted; on-chip powers	nonuniform film or poling (effective length < L, so η_norm falls with L); loss; quoted with fiber-coupled powers
%/W	P_2ω/P_ω² for one device	rings and resonators; also whole waveguides	compares devices of different length or Q; a ring value scales with resonant build-up
absolute %	P_2ω/P_ω at a stated pump	high-power, depleted regime	says nothing about low-power efficiency
pulse energy (fJ, pJ)	energy at which a stated conversion or gain is reached	femtosecond, dispersion-engineered devices	not convertible to CW numbers without the pulse shape

Ring build-up from coupled-mode theory, H/ħ = g(a²b^† + h.c.), energy decay rates κ_a (fundamental), κ_b (SH), external rates κ_a,ex, κ_b,ex, zero detuning derived:

\frac{P_{b}}{P_{a}^{2}} = \frac{64 g^{2} κ_{a, ex}^{2} κ_{b, ex} ω_{b}}{ħ ω_{a}^{2} κ_{a}^{4} κ_{b}^{2}} \overset{κ_{ex} = κ / 2}{\to} \frac{16 g^{2}}{ħ ω_{a} κ_{a}^{2} κ_{b}}

At fixed g a ring's %/W therefore scales as Q³. Published definitions of g differ by numerical factors, so g values from different papers are compared only after conversion to one Hamiltonian. η_norm scales as 1/(λ²A_eff) at fixed d_eff (eq. 2.13); A_eff itself scales roughly with λ² in a geometrically scaled guide, and d_eff disperses (Miller). Blue SHG at 33,000 %/W/cm² reported and 1550 nm SHG at 2600 to 9600 %/W/cm² are therefore not ranked by one number.

Overlap

For dominant field components normalized to unit power, the integral in the numerator runs where d is nonzero (LN, with sign flips in poled regions). Some authors fold n_eff into this definition.

\frac{1}{A_{eff}} = \frac{{| \int_{χ^{(2)}} E_{2 ω}^{*} E_{ω}^{2} d A |}^{2}}{{(\int {| E_{ω} |}^{2} d A)}^{2} \int {| E_{2 ω} |}^{2} d A}

Coupled-mode equations

|A|² = power, fields written as A exp(i(ωt − kz)), the complex conjugate of the convention in Boyd, so phases of A carry the opposite sign (Armstrong et al. 1962). The equations conserve P_ω + P_2ω (Manley-Rowe) and reproduce eq. 2.13 and the tanh² law; the SHG figure in L1 integrates them by RK4.

\frac{d A_{ω}}{d z} = - i κ A_{2 ω} A_{ω}^{*} e^{- i Δ k z}, \frac{d A_{2 ω}}{d z} = - i κ A_{ω}^{2} e^{i Δ k z}, κ^{2} = η_{norm}

Error sources in measured η_norm

Facet-loss calibration at two wavelengths (the SH often exits multimode); poling duty-cycle errors and missing domains (d_eff falls as sin πD; Fejer et al. 1992); film-thickness variation that moves the phase-matching wavelength along the device, which adapted poling compensates (Chen et al. 2024); photorefraction and green-induced infrared absorption at high SH power (chapter 04). The d scale carries the largest systematic uncertainty: 25 vs 34 pm/V changes η_norm by 1.85× derived.

Reviews: TFLN platform (Zhu et al. 2021); LN photonics from UV to microwave (Boes et al. 2023); LNOI phase-matching schemes (BPM, modal, QPM) and tuning effects (You et al. 2024).

Reported SHG efficiencies by dateone unit per panel

2018-11-07Wang et al. 20182600 %/W/cm²waveguide, normalizedreportedTFLN PPLN, 1.5 µm pump, 4 µm period

■ waveguide %/W/cm²□ DFG %/W/cm²▲ waveguide %/W◆ ring %/Wconverterring model

Show

y axes

η_norm 2600 %/W/cm²

Waveguide L 5.00 mm

Ring Q (both modes)1.00e+05

Ring g/2π template1.00 MHz

η_norm L² at the set L	650.0 %/W	derived
ring, critical coupling, Q_a = Q_b = Q	1.373e+05 %/W	derived
κ_a/2π = ν_a/Q (1550 nm)	1934.1 MHz	derived

Hover a marker for its record. Values: the L2 table, each as reported. Ring model: 1550 nm fundamental, critical coupling. The Q slider moves only the lower panel.

TFLN PPLN, 1.5 µm pump, 4 µm period reported

Drawnupper panel per watt and square centimetre, lower panel per watt for one device; a point never appears on a panel of another unit. Drag Q: the ring line climbs as Q³ past the ring records while the upper panel stays still. Marker fill: color of the generated light where the source gives the pump wavelength; IR output dashed. Values: the measured-efficiency table in L2, each as reported. Ring model: 1550 nm fundamental, critical coupling, Q_a = Q_b = Q; g/2π is a display parameter template.

records as reported reported converter and ring model derived gold rule: the 1993 field-poled waveguide reported

Show

Vertical scale

Normalized efficiency 2600 %/W/cm²

Waveguide length 5.00 mm

Ring quality factor Q1.00e+05

Ring coupling g/2π template1.00 MHz

Record: Wang et al. 2018
Issue date: 2018-11-07
Value: 2600 %/W/cm²
Normalization: waveguide, normalized
η_norm L² at the set L: 650.0 %/W
Ring model %/W (critical, Q_a = Q_b): 1.373e+05
κ_a/2π = ν_a/Q (1550 nm): 1934.1 MHz

References

26 sources cited in this section

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Boes et al. 2023	Lithium niobate photonics: Unlocking the electromagnetic spectrum. Science 379 eabj4396 (2023).	doi:10.1126/science.abj4396
Boyd & Kleinman 1968	Parametric Interaction of Focused Gaussian Light Beams. Journal of Applied Physics 39 3597-3639 (1968).	doi:10.1063/1.1656831
Boyd 2008	Nonlinear Optics, 3rd ed.. Academic Press (2008).	doi:10.1016/b978-0-12-369470-6.00014-9
Chang et al. 2016	Thin film wavelength converters for photonic integrated circuits. Optica 3 531 (2016).	doi:10.1364/OPTICA.3.000531
Chen et al. 2024	Adapted poling to break the nonlinear efficiency limit in nanophotonic lithium niobate waveguides. Nature Nanotechnology 19 44-50 (2024).	doi:10.1038/s41565-023-01525-w
Fejer et al. 1992	Quasi-phase-matched second harmonic generation: tuning and tolerances. IEEE Journal of Quantum Electronics 28 2631-2654 (1992).	doi:10.1109/3.161322
Jankowski et al. 2020	Ultrabroadband nonlinear optics in nanophotonic periodically poled lithium niobate waveguides. Optica 7 40 (2020).	doi:10.1364/OPTICA.7.000040
Kitamoto et al. 1995	Optical Review 2 280 (1995).	no DOI
Kleinman 1962	Nonlinear Dielectric Polarization in Optical Media. Physical Review 126 1977-1979 (1962).	doi:10.1103/PhysRev.126.1977
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