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<reference>
Roughness-Limited Performance in
Ultra-Low-Loss Lithium Niobate Cavities
A LI K HALATPOUR * , L UKE Q I , M ARTIN M. F EJER , AND A MIR H.
S AFAVI -N AEINI *

arXiv:2505.01913v3 [physics.optics] 20 May 2025

Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
* akhalat@stanford.edu, safavi@stanford.edu

Abstract: Achieving low optical loss is critical for scaling complex photonic systems. Thin-film
lithium niobate (TFLN) offers strong electro-optic and nonlinear properties in a compact platform,
making it ideal for quantum and nonlinear optics. While 𝑄 factors above 107 have been achieved,
they remain below the intrinsic material limit. We present a systematic study of scattering
losses due to roughness in TFLN racetrack cavities, isolating contributions from sidewall and
interface roughness. Quality factors up to 27 × 106 are demonstrated in waveguides with widths
of 2.2𝜆 (∼ 3.5 𝜇m), where interface roughness dominates, and up to 1.2 × 107 in narrower
waveguides 0.8𝜆 wide (∼ 1.2 𝜇m), where sidewall roughness is the primary limitation. Our
modeling framework, based on 3D wave simulations informed by AFM-measured roughness, is
material-independent and broadly applicable across integrated photonic platforms.
© 2025

1.

Introduction

As integrated photonic systems continue to scale in size and complexity, the demand for
material platforms that enable low-loss propagation while supporting both passive and active
functionalities becomes increasingly critical. Thin-film lithium niobate (TFLN) presents a
compelling solution, offering low optical loss over a broad spectral range—from the visible
to the mid-infrared—alongside strong electro-optic and second-order (𝜒 (2) ) nonlinearities [1].
These attributes make TFLN a highly versatile and promising material system for next-generation
integrated photonics [2].
While minimizing optical loss has always been central to system optimization, in emerging
applications such as optical squeezing [3, 4], single-photon processing [5], and quantum optical
computing [6], optical losses directly constrain system realization and yield. As a result,
understanding the mechanisms of loss in TFLN photonics and achieving ultra-high quality factors
has become essential for the feasibility of the next generation of photonic systems.
The intrinsic absorption of TFLN is estimated to be approximately 0.2 dB/m, corresponding
to an absorption-limited quality factor of ∼ 160 × 106 [7, 8]. However, fabrication-induced
scattering losses often dominate. Lithography processes introduce line-edge roughness which
can contribute to optical loss. Physical etching using high-energy argon ions—currently the
most effective method for TFLN—also induces sidewall roughness that increases scattering loss.
Ion bombardment can also lead to re-deposition of amorphous lithium niobate on the sidewalls,
requiring post-processing steps for removal. Standard cleaning agents such as HF (hydrofluoric
acid) or SC-1 (a mixture of ammonium hydroxide and hydrogen peroxide) can further roughen
the TFLN surface [9]. In addition, the material’s sensitivity to high-temperature processes poses
further fabrication challenges.
Fabricating high-quality factor devices across both low and high confinement regimes is
essential, as performance in these asymptotic limits helps isolate the contributions of sidewall
and interface scattering. This not only enables validation of modeling tools but also guides
the identification of optimal geometries for complex photonic structures. However, achieving
high-𝑄 in the low-mode-volume limit—where optical fields are concentrated near rough sur-

faces—remains challenging due to extreme sensitivity to fabrication imperfections, despite the
appeal of such devices for strong light-matter interactions and nonlinear effects [10]. In contrast,
high-mode-volume devices are less sensitive to fabrication variations and are primarily limited
by the intrinsic surface quality of TFLN. Although waveguides much wider than the optical
wavelength have demonstrated intrinsic quality factors as high as 29 × 106 [11], their utility is
limited by the onset of multimode behavior and reduced nonlinear interaction strength.
While previous studies have examined the role of surface roughness in TFLN propagation
loss [12, 13], a quantitative predictive tool is still lacking. A model that directly predicts
propagation loss due to roughness for any given waveguide geometry and surface roughness
would be valuable for optimizing TFLN-based devices. Previous approaches based on the LaceyPayne model (using the equivalent current method [14, 15]), originally developed for low-index
contrast waveguides, can produce order-of-magnitude errors for high-contrast waveguides [16]
and are therefore not suitable for TFLN waveguides. In this work, we employ a full-wave
simulation method to extract roughness-induced propagation loss from data obtained by atomic
force microscopy for the waveguide geometries of interest.
2.

Device Architecture Overview

In order to probe the contribution of sidewall roughness to the quality factor, we fabricate
racetrack cavities on both X- and Y-cut thin film lithium niobate on oxide wafers (LNOI). The
waveguide widths are varied between devices, as wider waveguides are expected to exhibit
reduced sidewall scattering loss. To minimize the impact of higher-order modes, we use a narrow
and single-mode feedline with a width of 1.2 𝜇m that is single side-coupled to our cavities. The
geometry of the racetracks and feedlines is shown in Figure 1(a).
In many applications, the thickness of TFLN needs to be adjusted from the thickness provided
by the manufacturer (NanoLN in our case). Therefore, to better understand the impact of interface
roughness, we prepared two sets of samples, one from X-cut wafers and the other from Y-cut
wafers. The X-cut wafers undergo a pre-thinning step using Ar bombardment to remove 200 nm
of TFLN, whereas the Y-cut wafer surfaces remain at the as-manufactured quality, achieved via
chemical mechanical polishing. Since both sets of waveguides have sidewalls oriented in the
𝑧-direction of the crystal, the impact of sidewall roughness is consistent across samples, providing

a

Pout

b

w
LN
SiO2

+Z
w

L

H

Lt

Lc

h

y

x=d/2
w0
300 µm

x=-d/2

x

Pin

Fig. 1. (a) Geometry of a racetrack and feedline, along with the cross-section of the
racetrack resonator characterized by (𝑤, 𝐻, ℎ), where 𝑤 is the waveguide width, 𝐻 is
the waveguide height, ℎ is the slab thickness, 𝐿 is the length of the straight section
of the racetrack, and 𝐿 t is the taper length (200 𝜇m). (b) Schematic illustration of a
waveguide with a rough boundary at 𝑥 = −𝑑/2.

an ideal platform to probe the influence of both top and bottom interface roughness.
3.

Modeling Methodology

3.1.

Roughness Modeling: Sidewalls

In this section, we derive an estimate of the power lost by the guided mode due to boundary
perturbations along the 𝑧-direction in a waveguide with its cross section in the (𝑥, 𝑦) plane. The
waveguide boundaries are at 𝑥 = −𝑑/2 and 𝑥 = 𝑑/2, and we assume that only the boundary at
𝑥 = −𝑑/2 is perturbed by roughness (see Figure 1(b)). Suppose that the dielectric tensor takes the
form 𝝐 (𝑥, 𝑦, 𝑧) = 𝝐 (𝑥, 𝑦) + 𝛿𝝐 (𝑥, 𝑦, 𝑧), where 𝝐 (𝑥, 𝑦) describes the ideal 𝑧-independent waveguide
and 𝛿𝝐 (𝑥, 𝑦, 𝑧) represents a weak 𝑧-dependent perturbation. The electric field E(𝑥, 𝑦, 𝑧) in a
waveguide with a dielectric tensor 𝝐 (𝑥, 𝑦, 𝑧) satisfies Maxwell’s equation
𝜔2 𝜇
𝝐 · E = 0.
𝑐2

(1)


𝜔2 𝜇
𝝐 · E + 𝛿𝝐 · E = 0.
2
𝑐

(2)

∇×∇×E+
We can rewrite (1) as:
∇×∇×E+
We write the total field as:

E(𝑥, 𝑦, 𝑧) = E0 (𝑥, 𝑦) 𝑒 𝑖𝛽𝑧 + E𝑟 (𝑥, 𝑦, 𝑧),
where E0 is a known modal field solution of the ideal waveguide and satisfies:
 𝜔2 𝜇
∇ × ∇ × E0 (𝑥, 𝑦) 𝑒 𝑖𝛽𝑧 + 2 𝝐 · E0 (𝑥, 𝑦) 𝑒 𝑖𝛽𝑧 = 0.
𝑐

(3)

Therefore, substituting E into equation (2) and neglecting second-order correction terms (i.e.,
𝛿𝝐 · E𝑟 ), we arrive at
∇ × ∇ × E𝑟 +

𝜔2 𝜇
𝜔2 𝜇
𝝐
·
E
=
−
𝛿𝝐 · E0 𝑒 𝑖𝛽𝑧 .
𝑟
𝑐2
𝑐2

(4)

Equation (4) can be interpreted as an equivalent problem in which the scattered fields, E𝑟 , are
generated by an equivalent current as
Jeq (𝑥, 𝑦, 𝑧) = − 𝑖 𝜔 𝛿𝝐 (𝑥, 𝑦, 𝑧) · E0 (𝑥, 𝑦) 𝑒 𝑖𝛽𝑧 .

(5)

This derivation corresponds to the so-called Born approximation [17]. The magnetic vector
potential corresponding to the equivalent source can be expressed as
∫ ∞∫ ∞∫ ∞
A(𝑥, 𝑦, 𝑥) =
Jeq (𝑥 ′ , 𝑦 ′ , 𝑧 ′ ) · G(𝑥 − 𝑥 ′ , 𝑦 − 𝑦 ′ , 𝑧 − 𝑧 ′ ) 𝑑𝑥 ′ 𝑑𝑦 ′ 𝑑𝑧 ′ ,
(6)
−∞

−∞

−∞

where G(r − hr′ ) is the dyadic Green’s
i function of the unperturbed background, which satisfies
𝜔2 𝜇
the equation ∇ × ∇ × − 𝑐2 𝝐 (𝑥, 𝑦) G(𝑥, 𝑦, 𝑧) = 𝛿(𝑥)𝛿(𝑦)𝛿(𝑧).
To proceed, we further assume that the roughness are perturbations along the sidewall that
appear as "bumps" that are uniform along the waveguide height (the 𝑦 direction). We are therefore
ignoring effects of sidewall angle (typically on the order of 10 degress). Hence, the roughness
depends only on 𝑧, and its variation is described by a small amplitude random function 𝑓 (𝑧),

which is small compared to the cross-sectional dimensions. Therefore, equation (6) can be further
simplified by using Taylor’s expansion as:
∫ ∞∫ ∞

A≈
𝑓 (𝑧 ′ ) Jeq (−𝑑/2, 𝑦 ′ , 𝑧 ′ ) · G 𝑥 + 𝑑/2, 𝑦 − 𝑦 ′ , 𝑧 − 𝑧 ′ 𝑑𝑦 ′ 𝑑𝑧 ′ .
(7)
−∞

−∞

This suggests that the source can be expressed as a convolution of a line source with a 𝑧-dependent
random function that modulates the phase, as follows:
n
o
Jeq (𝑥, 𝑦, 𝑧) = J 𝐿 (𝑦) ∗ 𝑒 𝑖𝛽𝑧 𝑓 (𝑧) , where J 𝐿 (𝑦) ≡ Jeq (−𝑑/2, 𝑦, 0) 𝛿(𝑥 + 𝑑/2) 𝛿(𝑧).
To calculate the far-field radiated power of the line source J 𝐿 (𝑥, 𝑦, 𝑧), we can write:
𝜔2 𝜇
𝝐 · A 𝐿 = 𝜇0 J 𝐿 .
(8)
𝑐2
We start with the magnetic vector potential A 𝐿 associated with a line source represented by the
current density 𝐽 𝐿 . Magnetic potential 𝐴(𝑥, 𝑦, 𝑧) can be expressed as a convolution integral:
∫ ∫ ∫
′
A(𝑥, 𝑦, 𝑧) =
A 𝐿 (𝑥 − 𝑥 ′ , 𝑦 − 𝑦 ′ , 𝑧 − 𝑧 ′ ) 𝑓 (𝑧 ′ )𝑒 𝑖𝛽𝑧 𝑑𝑥 ′ 𝑑𝑦 ′ 𝑑𝑧 ′ .
(9)
∇ × ∇ × A𝐿 +

The radiation modes in the Fourier domain lie on a surface of a sphere with radius 𝑘 = 𝑛clad 𝑘 0 .
Thus, the ensemble average of the intensity of radiated field in the far field is given by:
∫
1 1
⟨𝑃rad ⟩ =
⟨| Ê(k)| 2 ⟩ 𝑘 2 𝑑Ω,
(10)
(2𝜋) 2 2𝜂 |k|=𝑛clad 𝑘0
Here we used ⟨⟩ for ensemble averaging and 𝜂 for the wave impedance in the cladding material,
and the Ê(k) is the Fourier transform of the electric field. By employing the properties of
the convolution operator in the Fourier domain, and using the far-field approximation with the
relationship 𝐸 = −𝑖𝜔𝐴, the ensemble-averaged Fourier intensity is given by:
ˆ − 𝑘 𝑧 ),
⟨| Ê(k)| 2 ⟩ = 𝐿| Ê 𝐿 (k)| 2 𝑅(𝛽

(11)

where ˆ
𝑅(𝑘 𝑧 ) ≡ ⟨| 𝑓ˆ(𝑘 𝑧 )| 2 ⟩/𝐿 is the power spectral density of 𝑓 (𝑧), and Ê 𝐿 (k) represents the
Fourier transform components of the far field from the line source.
Therefore, we can write:
∫

⟨𝑃rad ⟩
𝑘2
2
=
Ê 𝐿 (k) 𝑅ˆ 𝛽 − 𝑘 𝑧 𝑑Ω.
𝐿
|k|=𝑛clad 𝑘0 2𝜂
2

(12)

2

𝑘
Defining 𝑆 𝐿 (𝜃, 𝜙) ≡ (2 1𝜋 ) 2 2𝜂
Ê 𝐿 (k) in (12) the far-field scattering power per unit length can
be approximated as
∫ 2𝜋 ∫ 𝜋


⟨𝑃rad ⟩
=
𝑆 𝐿 (𝜃, 𝜙) 𝑅ˆ 𝑛eff 𝑘 0 − 𝑛clad 𝑘 0 cos 𝜃 sin 𝜃 𝑑𝜃 𝑑𝜙,
(13)
𝐿
0
0

This yields the final expression for the scattering (or absorption) coefficient 𝛼𝑠 , which is defined
by the equation 𝑃rad /𝐿 = 𝛼𝑠 𝑃𝐺 , giving us:
∫ 2𝜋 ∫ 𝜋


1
𝛼𝑠 =
𝑆 𝐿 (𝜃, 𝜙) 𝑅ˆ 𝑛eff 𝑘 0 − 𝑛clad 𝑘 0 cos 𝜃 sin 𝜃 𝑑𝜃 𝑑𝜙,
(14)
𝑃𝐺 0
0
where 𝑃𝐺 is the guided mode power (i.e., the integral of the 𝑧-component of the Poynting
vector across the cross-section). If the roughness profiles for the left and right sidewall are
uncorrelated, then each sidewall contributes independently to the total scattering coefficient,
so the total is 𝛼𝑠tot = 𝛼𝑠(−𝑑/2) + 𝛼𝑠(+𝑑/2) . In the common special case where the sidewalls have
single sidewall
identical roughness statistics, this simply gives us 𝛼𝑠tot = 2𝛼𝑠
, giving an equation which
is identical to that used in [16].

3.2.

Roughness Modeling: Top and Bottom Surfaces

In ridge-waveguide geometries, scattering that arises from roughness at the top and bottom
interfaces can be described by a two-dimensional random function. Let 𝑓 (𝑥, 𝑧) denote isotropic
height variations at a rough interface at 𝑦 = 0. We assume 𝑓 (𝑥, 𝑧) is a zero-mean, stationary
random function with power spectral density (PSD) 𝑅ˆ 𝑓 (𝑘 𝑥 , 𝑘 𝑧 ). Following the same approach
used in (7), we can express the scattered magnetic vector potential A in the form:
∫ ∞∫ ∞

A(𝑥, 𝑦, 𝑧) ≈
𝑓 (𝑥 ′ , 𝑧 ′ ) Jeq (𝑥 ′ , 0, 𝑧 ′ ) G 𝑥 − 𝑥 ′ , 𝑦, 𝑧 − 𝑧 ′ d𝑥 ′ d𝑧 ′ ,
(15)
−∞

−∞

where Jeq (𝑥, 0, 𝑧) is the equivalent surface current of the fundamental mode. Here we approximate
the profile of the fundamental TE mode of a ridge waveguide as a cosine with length scale 𝑑 ′ in
the transverse direction. The surface current is given this cosine term modulating the roughness
function 𝑓 (𝑥, 𝑧) over a waveguide width 𝑑 as:
 𝜋𝑥 
𝑥
(16)
𝐽 (𝑥, 𝑧) = 𝑓 (𝑥, 𝑧) cos ′ rect
𝑒 𝑖𝛽𝑧 .
𝑑
𝑑
Here 𝑑 ′ is the extent of the mode (always larger than width, 𝑑, though 𝑑 ′ ≈ 𝑑 for wide waveguides),
and rect(𝑢) = 1 for |𝑢| ≤ 21 and zero otherwise. To find 𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) = ⟨| 𝐽ˆ(𝑘 𝑥 , 𝑘 𝑧 )| 2 ⟩, we start by
defining 𝑔(𝑥) where 𝐽 (𝑥, 𝑧) = 𝑓 (𝑥, 𝑧)𝑒 𝑖𝛽𝑧 𝑔(𝑥). Then:
∫ ∞
1
ˆ
𝑓ˆ(𝑘 ′𝑥 , 𝑘 𝑧 − 𝛽) ˆ𝑔(𝑘 𝑥 − 𝑘 ′𝑥 ) 𝑑𝑘 ′𝑥 .
(17)
𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) =
2𝜋 −∞
The power spectral density (PSD) 𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) of the random function 𝐽 (𝑥, 𝑧) is defined as the
ensemble average of the squared magnitude of its Fourier transform:
𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) =
𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) =

1
(2𝜋) 2

∫ ∞

| 𝐽ˆ(𝑘 𝑥 , 𝑘 𝑧 )| 2 .

2
𝑅ˆ 𝑓 (𝑘 ′𝑥 , 𝑘 𝑧 − 𝛽) ˆ𝑔(𝑘 𝑥 − 𝑘 ′𝑥 ) 𝑑𝑘 ′𝑥 .

(18)
(19)

−∞

The Fourier transform of g(x) can be expressed as:
ˆ𝑔(𝑘 𝑥 ) =




i
𝑑h
sinc 𝑑2 𝑘 𝑥 − 𝑑𝜋′ + sinc 𝑑2 𝑘 𝑥 + 𝑑𝜋′ .
2

(20)

In (20) we have 𝑑 ′ > 𝑑, so the two sinc lobes overlap slightly. The peaks are separated by
2𝜋/𝑑 ′ , whereas each main lobe has a width of order 2𝜋/𝑑; hence the lobes are not completely
isolated, and their overlap introduces a small cross term. Only in the limit 𝑑 ′ → 𝑑 (i.e., for very
wide waveguides) does the peak of one sinc nearly align with the first zero of the other, thereby
suppressing interference. Throughout the remainder of the analysis we neglect this cross term
and, for simplicity, replace each |sinc| 2 factor by a delta function of equal area. This yields:


1 𝑑 ˆ
𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) ≈
𝑅 𝑓 𝑘 𝑥 − 𝑑𝜋′ , 𝑘 𝑧 − 𝛽 + 𝑅ˆ 𝑓 𝑘 𝑥 + 𝑑𝜋′ , 𝑘 𝑧 − 𝛽 .
2𝜋 4
The scattering loss can then be calculated similarly as:
∫ 2𝜋 ∫ 𝜋
1
𝑆 𝑝 (𝜃, 𝜙) 𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 ) sin 𝜃 𝑑𝜃 𝑑𝜙,
𝛼𝑠 =
𝑃𝐺 0
0

(21)

(22)

where 𝑆 𝑝 (𝜃, 𝜙) is the power radiated per unit solid angle for a delta-function source defined as
J 𝑝 = J0 𝛿(𝑥) 𝛿(𝑦) 𝛿(𝑧). Here, J0 corresponds to the equivalent current at E0 at 𝑥 = 𝑦 = 𝑧 = 0.

Since the radiation intensity of a subwavelength radiator scales with its physical length squared
(i.e., ∝ 𝐿 2 for 𝐿 ≪ 𝜆), equation (22) can be rewritten in the following form:
∫ 2𝜋 ∫ 𝜋


𝑑
𝛼𝑠 =
𝐿𝑐



1
𝐿𝑐

𝑆patch (𝜃, 𝜙) 𝑅ˆ 𝐽 (𝑘 𝑥 , 𝑘 𝑧 )/(𝑑𝐿 2𝑐 ) sin 𝜃 𝑑𝜃 𝑑𝜙


0

0

,
𝑃𝐺

(23)

where 𝑆patch (𝜃, 𝜙) is the far-field radiation from a coherent patch of size 𝐿 𝑐 × 𝐿 𝑐 with uniform
current 𝐽0 . This form highlights that the prefactor represents the number of statistically independent scattering patches, interpreting roughness-induced loss as the sum of their uncorrelated
radiation.
3.3.

From propagation loss to intrinsic quality factor of a resonator

Throughout the remainder of the paper we quote and compare quality factors, because 𝑄 is what
we extract directly from the optical linewidth. Here we connect these measured quantities to the
scattering–loss coefficient 𝛼𝑠 that comes from the roughness model. Our on-chip power decays
as 𝑃(𝑧) = 𝑃0 𝑒 − 𝛼𝑠 𝑧 from which we obtain
𝑄i =

𝜔0
.
𝑣𝑔 𝛼

Generally, the propagation loss extracted from the scattering model we use does not translate
uniquely into a single quality factor. This is because the scattering may be into higher-order
modes of the waveguide. These modes may themselves resonate inside the cavity. The net effect
of this scattering can then be enhanced or suppressed depending on the detuning of the mode
of interest from the higher-order resonances of the cavity. In our case, we are assuming that
the sidewall and interface loss are effectively irreversible. This may be a good assumption if,
for example, there is enhanced out-coupling, scattering, and bend loss in higher-order modes.
Then, the energy scattered into a higher mode can be assumed to be effectively lost, and so these
feedback effects are not taken into account. In this work, we use the simple relation between 𝑄
and 𝛼 described above and ignore these effects.
4.

Measurements and Discussion

As shown in our theoretical work, our approach requires the far-field radiation pattern of a small
patch or a line source, along with the roughness parameters, to predict the scattering loss and
infer the quality factor of the resonator. The details of the 3D simulation, which we performed
using a 3D FDTD solver (Tidy3D) are provided in the Supplemental Material. In addition, due
to the discontinuity of electric fields at the sidewall for the fundamental TE mode, we used the
electrostatic limit and polarizability matrix to extract the equivalent current [18], as detailed in
our Supplemental Material. In the following section, we proceed with the extraction of roughness
parameters.
4.1.

Evaluating the Roughness by Atomic Force Microscopy

We performed atomic force microscopy (AFM) in tapping mode using an Asylum Research AFM
and a Bruker TESPA-V2-SS cantilever tip. To extract the roughness parameters, we conducted
two-dimensional scans over areas of 2–5 𝜇m × 5 𝜇m, covering both the sidewall and the top
surface of the waveguides. Figure 2 presents the autocorrelation of the 𝑍 + and 𝑍 − surfaces
extracted from AFM scans of a device with width 𝑤 = 1.2 𝜇m. The two lines represent the range
of the correlation length 𝐿 𝑐 . The insets show a top-view AFM scan and the sidewall power
spectral density (PSD). Although the sidewall PSD does not follow a purely Lorentzian profile,
it exhibits a slowly varying region with a kink at higher frequencies (corresponding to 1/𝐿 𝑐 ),

Z+ sidewall

10-

top surface

10-

10-

10-

Autocorrelation

1

10
10-

10-

Z- sidewall

Spatial Frequency (1/nm)
Lc = 40nm
Lc = 60nm

0
-1

b
PSD (nm )

10

Autocorrelation

PSD (nm )

a

0

10-

Spatial Frequency (1/nm)
1
Lc = 50nm
Lc = 70nm

0

1

τ (µm)

10-

-1

0

τ (µm)

1

Fig. 2. Autocorrelation function of the roughness obtained from AFM measurements
on the 𝑍 + and 𝑍 − sides of a lithium niobate waveguide with width 𝑤 = 1.2 𝜇m. The
two lines mark the range of the correlation length 𝐿 𝑐 . The insets show (i) a 3D AFM
image indicating line-like roughness features propagating through the sidewall and (ii)
the sidewall PSD with a corner frequency, validating the scan length and resolution.

confirming that the scan length and resolution were sufficient to capture the corner frequency. We
approximated 𝐿 𝑐 by analyzing the autocorrelation of the scan lines. The 3D AFM perspective
(see Fig. 2) indicates that the roughness appears as line-like features that propagate along the
sidewall, supporting a modeling approach that neglects variations along the waveguide height.
Table 1 summarizes the roughness parameters (𝜎) and correlation lengths (𝐿 𝑐 ) for devices with
widths of 1.2, 2.5, and 3.5 𝜇m. The 𝑍 + surface shows significantly higher roughness than the 𝑍 −
surface. Wider devices exhibit increased sidewall roughness, likely due to overexposure of the
lithography mask. This can be mitigated by optimizing the exposure dose for larger waveguide
widths. Due to the extremely smooth surface, tapping mode is not appropriate for measuring
the roughness of the top surface. We performed a ScanAsyst scan (peak force mode) for those
surfaces, and a detailed section is presented in the supplemental file.
Table 1. Roughness parameters 𝜎 and correlation length 𝐿 𝑐 for different sidewall
widths.

Width (𝜇m)

𝜎𝑧 + (nm)
𝜎𝑧 − (nm)

4.2.

1.2

2.5

3.5

Lc (nm)

1.4–1.6
0.4–0.5

2.2–2.4
0.4–0.5

2.2–2.45
0.4–0.5

40–60
45–65

Quality Factor Measurements

The experimental data, along with comparisons to simulated values, are shown in Figure 3.
Details concerning the extraction of quality factors from measured transmission spectra, as well
as the statistical parameters characterizing the top interface, are provided in the Supplemental
Material. Here, we decompose the quality factor into a width-dependent contribution from the

sidewall and a nearly constant term, according to
1
1
1
1
1
1
1
=
+
+
+
=
+ ,
𝑄 𝑄 sidewall 𝑄 top 𝑄 bottom 𝑄 absorption 𝑄 sidewall 𝐶

(24)

Our scattering model analysis which combines the AFM-derived roughness parameters for the
top, bottom, and independently measured bulk absorption [8] lead us to a best estimate for 1/𝐶
of 42–64 × 106 (see Supplementary Materials). We draw two observations from the data. First,
both Ar-trimmed and CMP-trimmed surfaces yield similar quality factors for narrow waveguides
(on the order of 107 ), in agreement with our model. This confirms that in narrow devices, losses
are dominated by roughness of the sidewall. In contrast, CMP-polished surfaces exhibit higher
average 𝑄 values in larger devices, where the top and bottom interface losses become more
significant. The superior surface quality achieved through CMP, as measured in this work and
reported in the literature [9], supports this conclusion. It should be noted that the final surface
quality of these waveguides is ultimately influenced by subsequent chemical processing. However,
as shown in the Supplemental Material, CMP-polished surfaces still provide interfaces superior
to those of Ar-trimmed samples, even after identical chemical treatments.

a ×106
Quality factor

60
50

1/C

40



σ1.2 um
σ3.5 um

30
20

CMP + chemical cleaning

10

Ar ion + chemical cleaning
1.2

Transmission (V)

b
0.3
0.2
0.1

2.5

1.2 µm width
0.4

0

1544

3.5

Waveguide width (µm)

Qt=9.5
Qi=12.3
1545

1544.6 nm

c
Transmission (V)

0

3.5 µm width

0.3
0.2
0.1
0

Qt=19.1
Qi=27.5

0.4
0
1531

1532

1531.4 nm

Fig. 3. (a) Comparison of experimentally measured 𝑄 values (boxplots) and simulated
𝑄 for waveguides with 𝐻 = 300 nm, ℎ = 200 nm, and widths 𝑤 = 1.2 𝜇m, 2.5 𝜇m, and
3.5 𝜇m. The black lines indicate the estimated bounds for the constant term 1/𝐶 [see
Eqs. (24)]. The gray shaded region shows the total expected loss including intrinsic,
interface, and sidewall contributions for the 1.2 𝜇m waveguide roughness, and the
area is the uncertainty in the AFM measurement in Table 1. Similarly the yellow
region corresponds to the 3.5 𝜇m waveguide roughness. (b) Transmission response
and corresponding measured 𝑄 for the single-mode 1.2 𝜇m device. (c) Transmission
response and measured 𝑄 for the single-mode 3.5 𝜇m device.

A similar observation was reported in Ref. [11], where 𝑄 ≈ 2.9 × 107 was achieved only in
very wide waveguides (𝑤 = 4.5 𝜇m, 𝐿 = 10 mm). By comparison, our results demonstrate that

comparable 𝑄 values can be attained in narrower waveguides, which are more attractive due to
their larger nonlinear and electro-optic coupling.
In general, these experiments show that for narrow waveguide resonators, losses are dominated
by sidewall roughness, which scales as 𝜎 2 . Minimizing edge roughness through improved
fabrication is crucial for achieving ultra-high-𝑄 devices. In this regime, our model predicts the
observed 𝑄 values, and points out at what level of performance the top and bottom surfaces will
begin to dominate quality factors.
5.

Concluding remarks

A comprehensive understanding of loss mechanisms in TFLN waveguides is essential for
continued progress in the field of nonlinear and quantum photonics. Our results provide a
quantitative framework for estimating the relative contributions of scattering mechanisms. The
methodology we have demonstrated is material-agnostic and more broadly applicable across
the integrated photonics community. Importantly, this study informs us of which aspects of
the fabrication process are the most fruitful to focus on for rapid progress. By systematically
identifying and addressing these losses, we believe that realizing the material limits of TFLN in
real devices is possible, and will lead to unprecedented device performance and new capabilities.
6.

Acknlowledgements

This work was supported by the U.S. government through the Defense Advanced Research
Projects Agency (DARPA) INSPIRED program, the National Science Foundation NSF-SNSF
MOLINO project No. ECCS-2402483, and the US Department of Energy through grant no.
DE-AC02-76SF00515 and via the Q-NEXT Center. Device fabrication was performed at
the Stanford Nano Shared Facilities (SNSF) and the Stanford Nanofabrication Facility (SNF),
supported by the National Science Foundation under award ECCS-2026822. This material is
based upon work supported by the National Science Foundation Graduate Research Fellowship
Program under Grant No. (DGE-1656518). L.Q. also gratefully acknowledges support from the
Shoucheng Zhang Graduate Fellowship Program. The authors appreciate financial and technical
support from NTT Research.
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7.

Supplemental Material

7.1.

Crystal Axis Dependent Etch in LN

As shown in the main text, the impact of sidewall roughness can be reduced by increasing the
waveguide width. We also observed that for narrower devices (𝑤 ≈ 1.2 𝜇m), the dominant
scattering arises from the sidewalls, whereas for wider devices (𝑤 ≈ 2.5 𝜇m), the primary
contribution to loss comes from the top and bottom interfaces.
Periodically poled lithium niobate (PPLN) waveguides (e.g., [19–21]) are of interest for
many applications. In PPLN, the optical axis of LN (+𝑍) is periodically flipped by applying a
strong electric field of approximately 3 V/𝜇m. This poling process can artificially increase the
crystal-dependent, fabrication-induced roughness, because the chemical processing used here
has different etch rates and roughness profiles on the +𝑍 and −𝑍 faces of LN.
Since the scattering loss of LN in narrow waveguides is dominated by sidewall roughness,
we expect that any additional roughness from poling will lead to pronounced loss in narrower
devices, while wider devices should be comparatively unaffected. To explore this effect, we
constructed a total length racetrack resonator 2.5 mm in which one arm (straight section) was
positioned, representing approximately 40% of the total cavity length. We tested two devices
on the same chip, separated by 550 𝜇m to minimise processing variations. One device had
a waveguide width of 𝑤 = 1.2 𝜇m, and another was tapered to 𝑤 = 2.5 𝜇m. We poled one
device, using the neighboring (non-poled) device as a baseline. Aside from this difference,
all fabrication steps were identical—including exposure to chemicals, solvents, and crucial
sidewall cleaning processes. Any observed differences are therefore attributed directly to these
crystal-axis-dependent fabrication processes. The experimental setup is illustrated in Figure 4.
Pout

y
z

x
-Z
Z

Ef=3 V/um
Tpoll

L

Lc

L

Lc

electrodes

Pin

30 µm
550 µm

Fig. 4. Schematic of our experimental setup, where the false-colored section represents
the poled region in a 2 mm racetrack. An SHG microscope image of the domain
formation with 𝑇pol = 3.7 𝜇m is shown alongside a schematic for clarity. An electric
field is applied to periodically change the crystal optical axis (𝑧). The darker lines
represent the poling fingers (aluminum), and the brighter areas show the depth of the
poled region in the slab. The poling region is 30 𝜇m wide and extends uniformly across
the waveguide width. Alignment marks ensure the waveguide is centered in the poled
slab.

As shown in Figure 5, the narrow device experiences a degradation of more than 30%, whereas

for the wider device the degradation is less than 7%, consistent with our expectations. A more
quantitative analysis of the poled devices will be presented in a separate study.
avg Qi, nonpoled = 2.5 x 106
avg Qi, poled = 1.8 x 106

1

0.6

0.2
1498

1502

1506

Wavelength (nm)

1510

b
Normalized transmisson

Normalized transmisson

a

avg Qi, nonpoled = 9.1 x 106
avg Qi, poled = 8.5 x 106

1
0.6
0.2
1498

1502

1506

1510

Wavelength (nm)

Fig. 5. Quality factors of racetrack resonators with 𝑤 = 1.2 𝜇m and 𝑤 = 2.5 𝜇m before
and after poling. The narrower device (left) exhibits more than a 30% degradation,
whereas the wider device (right) experiences less than a 7% reduction in quality factor.
This result confirms that wider devices are predominantly affected by top and bottom
interfaces, as derived in the main text.

7.2.

The effect of field discontinuity and large index contrast

For a bump at waveguide boundaries with a surrounding dielectric, discontinuities in the normal
component of E0 at the interface makes the product 𝛿𝝐 · E0 ill-defined [22]. One strategy to
mitigate this issue is to note that if the characteristic lengths of roughness are much smaller than
the wavelength, the scattering from this perturbation can be approximated by the long-wavelength
limit, for which the quasi-static approximation applies [18].
We now restrict our analysis to a waveguide with the permittivity tensor in matrix form given
by:
2
0ª
©𝑛 𝑒 0
­
®
𝜺 = 𝜖0 ­­ 0 𝑛2𝑜 0 ®® ,
­
®
2
« 0 0 𝑛𝑜 ¬
where 𝑛𝑒 is the extraordinary index (along ˆ𝑥) and 𝑛𝑜 is the ordinary index (along 𝑦ˆ and 𝑧ˆ), and
with 𝑛clad for the surrounding material. This corresponds to the LN system we are considering
where we use an X-cut TFLN film with waveguides oriented along the the crystal Y axis, so the Z
axis is transverse to the waveguide propagation axis. Then using [18], the following expressions
can be obtained (note that we’ve redefined the coordinate system so 0 is centered around the
sidewall at 𝑥 = −𝑑/2):


1 2
𝑛𝑜 + 𝑛2clad 𝛼 𝐸 ∥ (𝑥 = 0, 𝑦) 𝛿(𝑥) 𝛿(𝑧),
2
J⊥𝐿 (𝑦) = −𝑖 𝜔 𝜀 0 𝑛2clad 𝛾 𝐸 ⊥ (𝑥 = 0− , 𝑦) 𝛿(𝑥) 𝛿(𝑧).

J 𝐿∥ (𝑦) = −𝑖 𝜔 𝜀 0

(25)

Here, 𝐸 ⊥ (𝑥 = 0− , 𝑦) represents the field at the interface inside the cladding, while 𝐸 ⊥ (𝑥 = 0+ , 𝑦)
represents the field at the interface inside the waveguide. The currents 𝐽 𝐿∥ and 𝐽 𝐿⊥ denote the
tangential and normal induced currents at the rough interface, respectively. The tangential current
can be placed at the interface, whereas the normal current is located at 𝑥 = 0− , immediately
inside the cladding material. The polarizability coefficients 𝛼 and 𝛾 for a 2D bump can be
obtained from [18]







𝜏−1
𝜏−1
𝜏 − 1 
2(𝜏 − 1) 

.
1+ 𝜏
1+ 𝜏
, 𝛾(𝜏) =
(26)
𝛼(𝜏) =

1 
1 
𝜏 + 1 
𝜏
−
−
𝛼∞
𝛼0
𝛾∞
𝛾0



−1
−1
−1
−1
2
2




2
2
Here, the fitted parameters are 𝛼∞ = 0.8510, 𝛼0 = 3.882, 𝛾∞ = 3.905, and 𝛾0 = 0.7669. We
𝑛2
𝑛2
define 𝜏𝑜 = 𝑛2𝑜 and 𝜏𝑒 = 𝑛2𝑒 . For the sidewall, 𝛾(𝜏𝑒 ) is used for 𝐸 𝑥 and 𝛼(𝜏𝑜 ) is used for 𝐸 𝑦
clad

clad

and 𝐸 𝑧 . In this formulation, in the limit as 𝜏 → 1, the standard volume current (i.e., 𝐽 ∼ 𝛿𝝐 E) is
recovered.
For the upper and lower interfaces, analogous to the sidewall case, the polarizability matrix for
a 3D bump can be obtained from [18]. Here we consider only the tangential components, with
parameters 𝛼∞ = 1.22 and 𝛼0 = 2.85.
7.3.

Simulation Results of High-Index-Contrast Waveguides: Silicon on Oxide

To simulate the radiation profile of the equivalent line source, we used Tidy3D, a GPU-accelerated
platform. The calculation domain in each dimension spans approximately 10𝜆 to emulate the far
field, and the tests confirmed that further increases in the domain size do not alter the results. A
field monitor is placed to calculate the radiation intensity in the far field required for 𝑆 𝐿 (𝜃, 𝜙)
calculation in equation (14). The line source is placed at the boundary described in the modelling
section, and mesh refinement boxes are defined at the boundary to improve the accuracy of the
source placement. An important step in the simulation is to place a mode source to subtract
the radiation that couples back to the fundamental mode of the waveguide. The custom current
source feature of Tidy3D allows for the direct definition of a current source from modal analysis.
The setup is illustrated in the inset of Figure 6. To validate our simulation, we compared our
results with experimental data for a silicon-on-oxide waveguide, which features a high index
contrast and provides a robust data set for validation. Figure 6 shows the measurement results
from [23] along with our simulated values.

Loss (dB/cm)

14

Measured
Simulation

PML
Field monitor
Line source
Waveguide

10

Mode monitor

6
2
0.25

0.3

0.35

0.4

0.45

Waveguide Width (µm)

0.5

Fig. 6. Cladded SOI waveguide simulation with 𝐻 = 0.22 𝜇m, 𝜎 = 2 nm, and
𝑙 𝑐 = 50 nm (adapted from [23]). The inset shows the calculation domain in Tidy3D

7.4.

Statistical Parameters of the TFLN Interface

We used a Bruker Dimension Icon AFM equipped with a ScanAsyst-Air cantilever operating
in PeakForce mode for high-resolution surface characterization. To obtain representative,

artifact-free statistics, we also fabricated a reference sample of identical chip size and chemical
processing but without any patterning.
Figure 7 presents AFM scans of a sample that was CMP-polished, followed by HF/Piranha
cleaning and annealing. Although this surface is denoted “CMP” in the main text, we note that
the post-fabrication chemical steps dominate the final surface morphology.
Across three locations on the chip, we measured an RMS roughness of ∼ 180–200 pm and
a correlation length of ∼ 80–90 nm. The data were processed in Gwyddion: mean-plane
subtraction was applied, then median-based row alignment was used to suppress scan-line
artifacts.

a

0.0 µm
0.0

0.2

0.4

b
1.85 nm

0.0 µm
0.0

0.2

0.4

1.85 nm
1.50

1.50

1.00

0.50

0.4

c 0 µm
0

1

0.00

1

2

0.2

d

1.96 nm
1.50

1.00
2
0.50
0.00

1.00

0.50

0.4

Autocorrelation

0.2

0.00

1
0.5
0
-0.5

-2

0
Lag (m)

2
10

-6

Fig. 7. AFM analysis of a CMP-polished, chemically treated, and annealed TFLN
surface. (a) 0.5 × 0.5 𝜇m2 scan (1024 lines) showing a smooth surface with polishing
trenches. (b) Scan of a different region on the same sample with fewer polishing marks.
(c) 3 × 3 𝜇m2 scan (1024 lines) used to extract the correlation length. (d) Normalized
autocorrelation functions for 3 × 3 𝜇m2 (blue) and 5 × 5 𝜇m2 (red) areas, both decaying
to 1/𝑒 at ∼ 80–90 nm. The 5 × 5 𝜇m2 scan has ∼ 2048 × 2048 points.

Figure 8 shows AFM scans acquired with the same instrument and settings for a sample that
underwent neutral-argon-atom bombardment for trimming, followed by an SC-1 clean to remove
argon implantation from the top surface. The surface exhibits a different morphology, with no
polishing trenches; however, the RMS roughness increases to ∼ 200–220 pm. The correlation
length remains ∼ 100 nm, although some long-range features are visible in the scan.
Although we have made every effort to obtain reliable AFM data, we acknowledge that when
the surface is extremely smooth and atomic terraces are absent, the estimated roughness and
correlation length carry significant uncertainty. In our experience, the choice of leveling procedure
can also introduce an additional 10–30 pm variation in these parameters. To account for this,
we adopt conservative bounds for the TFLN statistical parameters, namely 𝜎 = 0.20–0.25 nm
and 𝐿 𝑐 = 100–150 nm. We also acknowledge that our assumption of isotropic roughness is

0.0 µm
0.0

0.2

b

0.4
1.97 nm

1.50
0.2
1.00

1

Autocorrelation

a

0.5

0

0.50

0.4

-2

0.00

0

2

Lag (m)

10

-6

Fig. 8. AFM analysis of the argon-milled, SC-1-cleaned TFLN surface. (a) 0.5×0.5 𝜇m2
scan (1024 lines) showing a smooth surface without polishing trenches. (b) Normalized
autocorrelation function from a 5 × 5 𝜇m2 scan, decaying to 1/𝑒 at ∼ 100 nm.

approximate.
In the following section we estimate the 𝑄 factor using two representative lower-bound cases:
(i) 𝜎 = 0.2 nm with 𝐿 𝑐 = 100 nm and (ii) 𝜎 = 0.25 nm with 𝐿 𝑐 = 150 nm. These same bounds
are used in the main-text evaluation of interface roughness.
It is worth noting that the optimized Sidewall cleaning protocol developed in this work
was recently employed in Ref. [9], where—under ideal annealing conditions—atomic-terrace
formation was demonstrated, suggesting that surface roughness below 0.2 nm is attainable.
Further investigation in this direction is left for future work.
7.5.

Simulation Results of High-Index-Contrast Waveguides: TFLN

By neglecting bending and coupling loss, the total loss in the racetrack resonators can be
decomposed into a sidewall-scattering term and a constant term that accounts for the top- and
bottom-interface losses together with the intrinsic absorption of lithium niobate:
1
1
1
=
+ ,
(27)
𝑄 𝑄 sidewall 𝐶
where 1/𝐶 represents the combined interface and intrinsic material losses. Because the
interface contribution is only weakly width-dependent for wide waveguides, we take 1/𝐶 from
simulations performed on a waveguide with 𝑤 = 3.5 𝜇m.
The ridge waveguide considered here has height 𝐻 = 300 nm and slab thickness ℎ = 200 nm.
Absorption loss values are taken from [8]. As discussed in the AFM section of this supplemental
material, we used two bounding cases for the top interface statistical parameters: (i) 𝜎 = 0.20 nm,
𝐿 𝑐 = 100 nm, and (ii) 𝜎 = 0.25 nm, 𝐿 𝑐 = 150 nm.
Since we do not have access to the bottom interface, we assume that the bottom surface has the
statistical parameters, 𝜎 = 0.20 nm and 𝐿 𝑐 = 100–150 nm. This choice is arbitrary but consistent
with our CMP-polished TFLN roughness. In addition, we assumed there was no loss of ion
implantationation at the buried oxide and lithium niobate interface.
Based on these assumptions, we estimate:
𝑄 top ≈ (90–200) × 106 ,

𝑄 bottom ≈ (160–236) × 106 ,

Assuming 𝑄 absorption of about 160 × 106 , this yields:
1
≈ (42–64) × 106 .
𝐶

a

b

×106

×106

Qsidewall

102

102

Qub
Qlb

Q

Qsidewall

Simulation
Fit

101

1

1.5

2

2.5

3

3.5

101

1

1.5

Width (µm)

2

2.5

3

3.5

Width (µm)

Fig. 9. Simulated results for a TFLN waveguide with 𝐻 = 300 nm and ℎ = 200 nm. (a)
Simulated 𝑄 from sidewall contribution along with a power-law fit for 𝐿 𝑐 = 50 nm and
𝜎 = 1 nm. The fitted equation is 𝑄 sidewall = 15 (𝑤/1.2 𝜇m) 2.65 (1 nm/𝜎) 2 × 106 . (b)
Sidewall contribution (dashed line) and two total quality factor bounds (solid lines)
including the proposed range for 1/𝐶.

From Figure 9, we can determine the sidewall contribution for devices with 𝑤 = 3.5 𝜇m and
𝜎 = 1 nm to be 256 × 106 . For the average roughness measured by AFM of 𝜎 ≈ 1.7 nm, we can
scale the sidewall contribution to find 𝑄 sidewall ≈ 89 × 106 .
Using the bound for 1/𝐶, the simulated total quality factor is then in the range:
𝑄 total ≈ (29–37) × 106 ,
which is consistent with our measured data.
The asymptotic behavior of our measured quality factors around 𝑄 ≈ 29 × 106 is also consistent
with the results reported in [11], which used similar TFLN material provided by the same
manufacturing company, NanoLN. In addition, as derived from the main text, we ignore the
cross-terms for the top and bottom interfaces, meaning that we underestimate the loss contribution.
Therefore, we still believe our suggested bound is an upper limit for finite-width waveguides.
For devices 𝑤 = 1.2 𝜇m, the roughness of the sidewall is about 1.1 nm, yielding an estimated
𝑄 sidewall ≈ 14 × 106 , which results in total 𝑄 values of ≈ 11–12 × 106 that are also consistent
with the measured data.
7.6.

Fano resonance in single side-coupled resonators

We measure the transmission spectrum of racetrack resonators by coupling light into and out of
the chip through a feedline. The feedline features grating couplers at both the input and output,
designed for 1550 nm, and is side-coupled to the racetrack resonator. Sweeping the tunable
Santec TSL-570 diode laser allows us to probe a full spectrum of various modes.
To extract the intrinsic quality factor from this measurement, we model the system as a
single-mode cavity with resonance frequency 𝜔0 , intrinsic loss rate 𝜅𝑖 , and external coupling rate
𝜅 𝑒 . Using the standard coupled mode theory:

√
𝑑𝑎(𝑡)
𝜅𝑒 + 𝜅𝑖 
= − 𝑖𝜔0 +
𝑎(𝑡) + 𝜅 𝑒 𝑠in (𝑡),
𝑑𝑡
2
where 𝑠in (𝑡) is the input field. The output field is given by
√
𝑠out (𝑡) = 𝑠in (𝑡) − 𝜅 𝑒 𝑎(𝑡).

(28)

(29)

Assuming a harmonic time dependence ∼ 𝑒 −𝑖 𝜔𝑡 , the cavity amplitude in the frequency domain is
given by
√
𝜅𝑒
𝑎(𝜔) =
𝑠in (𝜔).
(30)
𝑖(𝜔 − 𝜔0 ) + 𝜅𝑒2+𝜅𝑖
Substituting this into the input–output relation leads to the transfer function:
𝐻 (𝜔) = 1 −

𝜅𝑒
.
𝑖(𝜔 − 𝜔0 ) + 𝜅𝑒2+𝜅𝑖

(31)

For quality-factor calculations, it is often more convenient to express the square modulus of the
transmission:
2
𝑄 𝑄 𝑒−1
2
,
(32)
|𝐻 (𝜔)| = bg 1 −
𝜔0
1 + 2𝑖𝑄 𝜔−
𝜔0
where bg can be expressed as a first-order polynomial and represent the background power
transmission . The intrinsic quality factor is then defined via
1
1
1
=
−
.
𝑄𝑖 𝑄 𝑄𝑒
However, to accurately determine the propagation loss in our optical system, it is crucial to
account for feedline loss and laser jitter at low scan speeds. Weak reflections in the feedline
facets lead to Fabry–Pérot modes that pair to the resonator modes and produce Fano resonances,
resulting in asymmetric line shapes [24]. Traditional symmetric Lorentzian fitting does not
capture these asymmetries [25]. To include the impact of the Fabry–Pérot response in our fitting,
we use a diameter correction method [26], modifying the transfer function as:
"
#
𝑄 𝑄ˆ 𝑒−1 𝑒 𝑖 𝜙
𝐻 (𝜔) = (1 + 𝜀) 1 −
(33)
𝜔0 .
1 + 2𝑖𝑄 𝜔−
𝜔0
Here, 𝜙 is a parameter that accounts for the asymmetry, and 𝜀 is a complex-valued function
with small magnitude that represents the reflection from the feedline facets (analogous to the
mismatch between line impedance and load in microwave systems). Since we do not have
phase-sensitive measurements for the transmission line in our optical system, we approximate
this form by taking its modulus squared.
|𝐻 (𝜔)| 2 = bg 1 −

𝑄 𝑄ˆ 𝑒−1 𝑒 𝑖 𝜙
𝜔0
1 + 2𝑖𝑄 𝜔−
𝜔0

2

,

(34)

where 𝜙 is a fitting parameter that captures the asymmetry. The intrinsic quality factor is then
given by
!
cos 𝜙0
1
1
=
−
.
(35)
𝑄𝑖 𝑄
𝑄ˆ 𝑒
Here, 𝜙0 is the zeroth-order term in 𝜙. Another way to interpret the DCM is to rewrite the transfer
function as
𝑚 2
|𝐻 (𝜔)| 2 = bg 1 − 𝑒 𝑖 𝜙 ·
,
(36)
1+𝑖Ω
2𝜅 𝑒
2(𝜔 − 𝜔0 )
, 𝜅 tot = 𝜅 𝑒 + 𝜅 𝑖 , and Ω =
.
where 𝑚 =
𝜅𝑖 + 𝜅𝑒
𝜅 tot
Rearranging yields a Fano-like form:
|𝐻 (𝜔)| 2 = bg ×

(Ω − 𝑚 sin 𝜙) 2 + (1 − 𝑚 cos 𝜙) 2
.
1 + Ω2

(37)

This expression is equivalent to a Fano shape with a bias term, where the asymmetry is
determined by 𝑚 sin 𝜙.
As the device quality factor increases, the measurement becomes more sensitive to the natural
jitter of the scanning laser, which degrades at slow scan speeds. The Santec TSL diode laser
used in this experiment has a motor to scan the output laser wavelength without any feedback,
therefore the actual wavelength output of the laser is not known without any measurement. To
characterize the wavelength of the laser, we use a fiber MZI with a known FSR of 325 MHz. The
characterization at various scanning rates is shown in Figure 10. As shown, the laser motor scan
speed is not constant, and the jitter becomes more apparent at slower scan speeds below 5 nm/s.
Therefore, all scans are taken at 5 nm/s at low optical input power.

MZI transmission signal (V)

1.8

2 nm/s
5 nm/s
20 nm/s

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1547.46

1547.461

1547.462

1547.463

1547.464

1547.465

Nominal wavelength (nm)
Fig. 10. MZI transmission signal for laser wavelength calibration

At these scan speeds, high-𝑄 resonances may only have a few data points per mode, which is
sufficient to extract the linewidth but can affect the fitting near the resonance minimum |𝐻 (0)|.
Since |𝐻 (0)| directly determines 𝑄 𝑖 from the measured linewidth, interpolation can be used to
improve data sampling, though it tends to very slightly decrease the extracted 𝑄 𝑖 , ensuring the
measured results are conservative estimates.
Figure 11 shows the Lorentzian and DCM fits for a representative mode from the main text.
As we focus on modes with low asymmetry, the fitted values are expected to be similar.
7.7.

Automatic fitting and symmetry filtering of modes

The procedure for mode fitting in the 𝑤 = 1.2 𝜇m device is elaborated in this section. We begin
by automatically detecting minima across the full transmission spectrum, as shown in Figure 12.
Next, we apply a symmetry-based filtering step by excluding modes with large asymmetry
(quantified by the phase parameter |𝜙0 | > 0.2) and poor fit quality. The result of this filtering is
shown in Figure 13, which includes approximately 20 modes shown in Figure 13.

Lorentzian without interpolation

a

0.6
0.4
0.2

0
1531.459

1531.46

1531.461

1531.462

1531.463

1

Transmission (V)

Qi=28.4×106

1531.464

0.8

0.4
0.2

0
1531.459

1531.465

Qi=28.1×106

0.6

1531.46

1531.461

Wavelength (nm)

c

d

DCM without interpolation

Qi=27.7×106

0.4
0.2

0
1531.459

1531.46

1531.461

1531.462

1531.463

1531.463

1531.464

1531.465

1531.464

1531.465

DCM with interpolation
1

Transmission (V)

Transmission (V)

0.8
0.6

1531.462

Wavelength (nm)

1

1531.464

1531.465

0.8

Qi=27.5×106

0.6
0.4
0.2

0
1531.459

1531.46

1531.461

1531.462

1531.463

Wavelength (nm)

Wavelength (nm)

Fig. 11. Fitting for Lorentzian and DCM method with and without interpolation.

1

Measured Transmission
Detected Minima

0.8

Transmission (V)

Transmission (V)

0.8

Lorentzian with interpolation

b

1

0.6

0.4

0.2

0

1525

1530

1535

1540 1545

1550 1555

Wavelength (nm)

1560 1565

Fig. 12. Automatically detected minima.

1570

1575

14

10 6

Qi
Q

Quality factor

12
10
8
6
4
2
0
0

5

10

15

20

Mode number

25

1545.114

1

1570.52

0.5

1544.683

Wavelength (nm)
1

1

0.5

0
1545.112

Wavelength (nm)

1
0.5

0

Wavelength (nm)

1

0

1537.144

1537.146

0
1572.303

1572.305

Wavelength (nm)

0.5

1566.725

1566.727

Wavelength (nm)

0.5

0
1570.518

1

0.5

0
1544.681

Transmission (norm)

Wavelength (nm)

Transmission (norm)

1532.985

0.5

1

Transmission (norm)

0
1532.983

1

Transmission (norm)

0.5

Transmission (norm)

1

Transmission (norm)

Transmission (norm)

Transmission (norm)

Transmission (norm)

Fig. 13. Filtered modes based on fit error and phase asymmetry 𝜙0 . Approximately 20
modes satisfy the criteria and are used for statistical analysis.

Wavelength (nm)

0.5

0
1572.642

1572.644

Wavelength (nm)

0
1547.491

1547.493

Wavelength (nm)

Fig. 14. DCM fitting results for the top 9 modes with the highest intrinsic quality factor
𝑄 𝑖 for the 𝑤 = 1.2 𝜇m device. Inset labels show extracted values of 𝑄 𝑖 and loaded 𝑄
for each mode.
</reference>

<statements>
1. The "Roughness-Limited Performance in Ultra-Low-Loss Lithium Niobate Cavities" study (arXiv:2505.01913, 2025) decomposes cavity loss into a sidewall-scattering term and a constant interface/absorption term, [arxiv] and shows that for narrow waveguides losses are dominated by sidewall roughness.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.