4.8 Thermal effects: thermo-optic shifts, thermal lenses, ring bistability

L1 Phenomenon

heat moves n_e more than n_o

Heat changes lithium niobate's extraordinary index 5 to 11 times more than its ordinary index, depending on wavelength and source (table below; thermo-optic coefficient). A ring resonator heated by its own light shifts to longer wavelength, so a laser sweeping upward drags the resonance with it and the transmission dip turns into a triangle (thermal triangle). A strong beam in a bulk crystal heats its own path and builds a lens.

Thermal calculator: dn/dT, thermal lens, ring shiftcongruent LN

dn/dT used3.75e−05 /K

thermal lens f7.4e+01 mmf/L 3.7; formula assumes f >> L

ring shift 20.2 pm/K

dn_e/dT at λ, T: 3.75e−05 /K
Moretti 632 nm e / o: outside 300-515 K
λ: 1.550 µm

Thesis example (10 W, w 150 µm, 0.005 cm⁻¹, 20 mm, 100 °C, 5 W/(m K)): 61 mm from its own formula with Moretti dn_e/dT at 632 nm; the thesis states about 100 mm. Which dn/dT it used is not stated.

Wavelength 1.550 µm

Temperature 25 °C

Lens: power 1.00e+01 W

Lens: radius w 150 µm

Lens: α 5.00e−03 /cm

Lens: length 20.0 mm

Ring: Γ 0.80

Ring: n_g 2.30

dn/dT for lens, ring

K_th, W/(m K)

Jundt dn_e/dT derived Moretti fits S 1523 nm value S dn_o/dT spot values from a vendor Sellmeier V K_th P-comp lens formula S Γ, n_g: model parameters, no data template Gayer 2008 MgO:LN excluded (anomalous dn/dT)

A crystal that becomes a lens

absorbed power heats the beam path; dn/dT > 0 focuses

Beam power 10.0 W

Absorption5.0e−03 /cm

Beam radius 150 µm

Crystal length20.0 mm

Wavelength1.550 µm

dn/dT for the lens and ring

Thermal conductivity, W/(m K)

Temperature 25 °C

Focal length: 7.4e+01 mm
f / L (formula assumes f >> L): 3.7
dn/dT used: 3.75e−05 /K
Source of dn/dT: e, Jundt at λ, T
Ring Γ, n_g (template): 0.80, 2.30

Thesis example (10 W, 150 µm, 0.005 cm⁻¹, 20 mm, 100 °C, 5 W/(m K)): 61 mm from its own formula with the Moretti dn_e/dT at 632 nm; the thesis states about 100 mm, and which dn/dT it used is not stated.

dn_e/dT, Jundt derivedMoretti fits S1523 nm value Sdn_o/dT spot values from a vendor Sellmeier Vlens formula SK_th P-compGayer 2008 MgO:LN excluded (anomalous dn/dT)ring Γ, n_g: model parameters template

L2 Coefficients, lens, ring shift

dn/dT by source

Quantity	Value	Conditions	Source
dn_e/dT	3.34×10⁻⁵ /K	1523 nm, 300 K	Moretti et al. 2005, via Zhu 2021S
dn_e/dT	3.75 / 4.01 / 4.89 ×10⁻⁵ /K	CLN, 1550 / 1064 / 633 nm, 25 °C; Jundt 1997 Sellmeier, recomputed	derived
dn_o/dT	0.41 / 0.54 / 0.94 ×10⁻⁵ /K	CLN, 1550 / 1064 / 633 nm, 25 °C; data-sheet Sellmeier, recomputed	Crystal Technology 1999derived from V
dn_o/dT, dn_e/dT at 632 nm	(−1.7 + 6.9×10⁻³ T)×10⁻⁵ /K; (−2.6 + 22.4×10⁻³ T)×10⁻⁵ /K; at 300 K: 0.37 and 4.12×10⁻⁵	T in K, 300-515 K	Schwesyg 2011, Eqs. 2.26-2.27 (Moretti 2005)S
Thermal conductivity	3.92 W/(m K) CLN; 5.97 W/(m K) near-SLN	room temperature	Volk & Wöhlecke 2009, Table 1.1P-comp
Not for dn/dT	Gayer 2008 MgO:LN coefficients give anomalous dn/dT (e.g. −7.1×10⁻⁵ /K for MgO1-SLN n_e at 1550 nm)		derived; excluded

Thermal lens (Schwesyg Eq. 2.28 S): f ≈ πw²K_th/(αPL dn/dT), valid for f ≫ L.
Ring thermal shift: dλ_res/dT = λΓ(dn_eff/dT)/n_g, with Γ and n_g from the mode (templates in the figure).
x-cut anisotropy: in x-cut TFLN the thermo-optic response depends on propagation angle and polarization through the mode's overlap with the o and e axes; a generalized analytical model exists (Shim et al. 2026reported).
Athermal ring: a thin TiO₂ cladding on an x-cut LN ring cancels the first-order thermo-optic coefficient at 20 °C; residual second order 0.37 pm/K²; resonance within 0.33 nm over −10 to 50 °C (Ling et al. 2020reported).
High power: absorption heating and beam-quality loss reported in MgO:PPLN above 10 W pump; an absorption of 10⁻⁴ cm⁻¹ already matters for cw singly resonant OPOs pumped at 10-20 W (Schwesyg p. 21 S).

Conflict. The thesis thermal-lens example (10 W, w = 150 µm, α = 0.005 cm⁻¹, L = 20 mm, 100 °C, K_th = 5 W/(m K)) gives f = 61 mm with dn_e/dT at 632 nm and 373 K from the Moretti fit derived; the thesis states about 100 mm. Same order; which dn/dT the thesis used is not stated.

L3 Bistability and sources

thermal triangle; dn/dT provenance

Thermal bistability (Carmon et al. 2004 reported (title)). Standard model used by the figure: absorbed power heats the ring, n rises, the resonance red-shifts; a laser swept toward longer wavelength drags the resonance along its flank until it drops off (triangular dip); swept toward shorter wavelength it jumps across the resonance (narrowed dip). Photorefraction in LN adds a slower term of opposite sign (Lin et al. 2021reported), with reported oscillation dynamics (Sun et al. 2017reported (title)) and self-starting soliton combs (He et al. 2019reported).
dn/dT sources: Moretti 2005 (320-515 K, 632 and 1523 nm, interferometric; abstract read, table not read); Jundt 1997 (temperature-dependent n_e Sellmeier; coefficients via a secondary). dn_o/dT disagrees by up to 2.5× between the data-sheet Sellmeier and the Moretti fit.
Measurement: interferometric phase versus temperature on a plate; ring resonance tracking versus stage temperature, separated from optical self-heating by low probe power.

Ring sweep with photorefraction off; switch it on to compare with 4.3. The template model shows the shortened triangle; it does not produce the oscillations reported by Sun et al.

Ring sweep: through-port T(λ), both directionssolid: λ increasing; dashed: decreasing

sweep: 1.00e+00 nm/s
P: 1.00e+00 mW
thermal shift, max: 1.01e+01 pm
PR shift, max: 0.00e+00 pm
T_min up / down: 0.000 / 0.376
Q_L, linewidth: 5.00e+05, 3.10 pm

Pump power 1.00e+00 mW

Sweep speed 1.00e+00 nm/s

PR strength 1.0 ×

Photorefraction

Cladding

dn_e/dT

Inset

dn_e/dT 1550 nm: Jundt 1997 Sellmeier, coefficients S, derivative derived or 1523 nm S r₃₃ 30.8 pm/V P-compn_e 2.1376 at 1550 nm P-db PR sign opposite to thermal reported cladding trend reported R 100 µm, Q_i 10⁶, n_g 2.3, Γ 0.8, R_th, τ_th, κ, σ_d, PR gain: model parameters, no data template

The thermal triangle

TFLN ring, radius 100 µm, 1550 nm laser swept across one resonance

Pump power1.0e+00 mW

Sweep speed1.0e+00 nm/s

Photorefraction

Photorefraction strength 1.0 ×

Cladding

dn_e/dT, 10⁻⁵ /K

Direction shown by the laser

Laser position in the sweep −8.1 pm

Transmission: 0.967
Thermal shift, max: +10.1 pm
Photorefractive shift, max: 0.0 pm
Loaded Q, linewidth: 5.00e+05, 3.10 pm

dn/dT 1550 nm, Jundt coefficients S, derivative derived or 1523 nm Sr₃₃ 30.8 pm/V P-compn_e 2.1376 at 1550 nm P-dbcladding trend reportedR 100 µm, Q_i 10⁶, n_g 2.3, Γ 0.8opposite-sign photorefraction reportedring, heat sinking, photoconductivity: model parameters, no data templatelight: 1550 nm, false color, dashed

Sources

Moretti et al. 2005L. Moretti, M. Iodice, F. G. Della Corte, I. Rendina, Temperature dependence of the thermo-optic coefficient of lithium niobate, from 300 to 515 K in the visible and infrared regions. Journal of Applied Physics 98 036101 (2005), doi:10.1063/1.1988987.Read: abstract (numbers via Zhu et al. 2021 and Schwesyg 2011, S)
Jundt 1997D. H. Jundt, Temperature-dependent Sellmeier equation for the index of refraction, n_e, in congruent lithium niobate. Optics Letters 22 1553 (1997), doi:10.1364/OL.22.001553.Read: not read; coefficients via a secondary source (S)
Crystal Technology 1999Crystal Technology Inc., LiNbO3 material properties, data sheet DC-12 8/99. (1999).Read: data sheet values via a secondary compilation
Zhu et al. 2021Di Zhu, Linbo Shao, Mengjie Yu et al., Integrated photonics on thin-film lithium niobate. Advances in Optics and Photonics 13 242 (2021), doi:10.1364/AOP.411024.Read: secondary use only
Carmon et al. 2004T. Carmon, L. Yang, K. J. Vahala, Dynamical thermal behavior and thermal self-stability of microcavities. Optics Express 12 4742 (2004), doi:10.1364/OPEX.12.004742.Read: title
Lin et al. 2021Z. Lin, Z. Kang, P. Xu, Y. Tian, S. He, Turnkey generation of Kerr soliton microcombs on thin-film lithium niobate on insulator microresonators powered by the photorefractive effect. Optics Express 29 42932 (2021), doi:10.1364/OE.446527.Read: abstract
Sun et al. 2017X. Sun, H. Liang, R. Luo, W. C. Jiang, X.-C. Zhang, Q. Lin, Nonlinear optical oscillation dynamics in high-Q lithium niobate microresonators. Optics Express 25 13504 (2017), doi:10.1364/OE.25.013504.Read: title
He et al. 2019Yang He, Qi-Fan Yang, Jingwei Ling et al., Self-starting bi-chromatic LiNbO3 soliton microcomb. Optica 6 1138 (2019), doi:10.1364/OPTICA.6.001138.Read: arXiv abstract
Ling et al. 2020J. Ling, Y. He, R. Luo, M. Li, H. Liang, Q. Lin, Athermal lithium niobate microresonator. Optics Express 28 21682 (2020), doi:10.1364/OE.398363.Read: abstract
Shim et al. 2026Joonsup Shim, Seonghun Kim, Shengyuan Lu et al., Generalized Model of Anisotropic Thermo-Optic Response on Thin-Film Lithium Niobate Platform. ACS Photonics 13 2586-2596 (2026), doi:10.1021/acsphotonics.6c00117.Read: arXiv abstract
Volk & Wöhlecke 2009Tatyana Volk, Manfred Wöhlecke, Lithium Niobate: Defects, Photorefraction and Ferroelectric Switching. Springer Series in Materials Science 115 (2009), doi:10.1007/978-3-540-70766-0.Read: selected chapters: Ch. 2.2 (pp. 14-19), Ch. 3 (pp. 51-61), Ch. 4 (pp. 75-77, 82-85, 94-97), Sec. 7.1.3 (p. 168)
Schwesyg 2011Judith R. M.-L. Schwesyg, Interaction of light with impurities in lithium niobate crystals. Dr. rer. nat. dissertation, Universität Bonn (2011).Read: selected: Sec. 2.2.2-2.3.2 (pp. 10-21), Ch. 5 Sec. 5.1 (pp. 69-72), Sec. 5.4-5.5 (pp. 99-101)