Quadratic Transmittance in Optical and Quantum Systems

• Quadratic transmittance is a phenomenon where transmission loss or phase modulation scales as a quadratic function of system parameters, such as beam misalignment or refractive-index mismatch.
• It manifests across diverse contexts including Gaussian beam optics, paraxial propagation with quadratic phase factors, anisotropic dispersion in 2D materials, and scattering in transparent ceramics.
• In quantum key distribution, quadratic transmittance underpins a rate scaling improvement from O(η) to O(√η), highlighting its practical impact on secure long-distance communication.

Quadratic transmittance denotes a family of context-dependent phenomena in which a transmittance, transmission-related observable, or transmission operator is controlled by a quadratic structure. In the cited literature, this structure appears in at least five technically distinct forms: the quadratic decrease of Gaussian-beam power transmission under small lateral misalignment, the quadratic phase factor $\exp\!\bigl(b_2 x^2/2\bigr)$ in paraxial propagation, the transmission anisotropy generated by a semi-Dirac dispersion that is quadratic in one momentum direction, the quadratic dependence of scattering and reflection losses on refractive-index mismatch in transparent ceramics, and the “quadratic improvement” of quantum-key-distribution rate scaling from $O(\eta)$ to $O(\sqrt{\eta})$ (Khwaja et al., 2016, Korneev et al., 2024, Nualpijit et al., 2017, Xiong et al., 30 Apr 2025, Fang et al., 2019).

1. Terminological scope and recurring mathematical forms

The expression does not denote a single universal law. In beam-aperture optics, it refers to the leading $d^2$ dependence of power transmittance for a laterally shifted Gaussian beam near perfect alignment. In paraxial Fourier optics, it denotes a quadratic multiplicative factor in the transverse coordinate, typically associated with thin lenses, quadratic phase plates, or the effective action of a quadratic-index medium. In anisotropic 2D materials, it refers to a quadratic electronic dispersion direction whose optical conductivity scaling controls polarization-dependent transmittance. In transparent piezoelectric ceramics, quadratic behavior enters through reflection and scattering terms proportional to $(\Delta n)^2$. In long-distance QKD, the phrase appears in the distinct sense of a quadratic improvement in the rate-transmittance relation, where secure key rate scales as $O(\sqrt{\eta})$ rather than $O(\eta)$ (Khwaja et al., 2016, Korneev et al., 2024, Nualpijit et al., 2017, Xiong et al., 30 Apr 2025, Fang et al., 2019).

A common mathematical motif is the suppression of linear terms by symmetry or factorization. For a centered circular aperture, the beam-displacement dependence satisfies $T(d)=T(-d)$, so the Taylor series begins at $d^2$. In paraxial propagation, Lie-algebraic decompositions isolate a quadratic multiplication operator $\exp(b_2 J_2)$, with $O(\eta)$0. In scattering theory, Fresnel intensity and RGD attenuation are quadratic in refractive-index mismatch. In semi-Dirac systems, the quadratic structure is not a Taylor coefficient of $O(\eta)$1 itself but a quadratic band curvature that enforces $O(\eta)$2. In QKD, the “quadratic” terminology concerns an exponent change in the transmittance scaling law rather than a quadratic polynomial (Korneev et al., 2024, Nualpijit et al., 2017, Xiong et al., 30 Apr 2025, Fang et al., 2019).

2. Quadratic power transmittance of a shifted Gaussian beam

For a fundamental TEM$O(\eta)$3 Gaussian beam incident on a circular aperture of radius $O(\eta)$4, with beam radius $O(\eta)$5 at the aperture plane and lateral displacement $O(\eta)$6 along the $O(\eta)$7-axis, the irradiance in aperture-centered cylindrical coordinates is written as

$O(\eta)$8

The transmitted power is

$O(\eta)$9

and the total beam power is

$O(\sqrt{\eta})$0

The power transmittance is therefore

$O(\sqrt{\eta})$1

Using the identity

$O(\sqrt{\eta})$2

the exact result becomes

$O(\sqrt{\eta})$3

with $O(\sqrt{\eta})$4 the lower incomplete Gamma function (Khwaja et al., 2016).

The on-axis case $O(\sqrt{\eta})$5 reduces to the standard Gaussian–circular-aperture formula

$O(\sqrt{\eta})$6

For small lateral shift, introducing $O(\sqrt{\eta})$7 and $O(\sqrt{\eta})$8, expansion to $O(\sqrt{\eta})$9 yields

$d^2$0

Equivalently, with $d^2$1 and $d^2$2,

$d^2$3

The linear term is absent because the configuration is symmetric under $d^2$4, so $d^2$5 is an even function. In this setting, “quadratic transmittance” denotes the leading behavior of the transmitted power fraction under small misalignment: the maximum occurs at $d^2$6, and the reduction is proportional to $d^2$7 (Khwaja et al., 2016).

The dimensionless coefficient

$d^2$8

controls curvature at the origin. For $d^2$9, $(\Delta n)^2$0 and $(\Delta n)^2$1, so transmission becomes insensitive to small shifts. For $(\Delta n)^2$2, $(\Delta n)^2$3, while the absolute transmitted power is already small. The data indicate that sensitivity peaks at intermediate aperture-to-beam ratios, since $(\Delta n)^2$4 has a maximum at finite $(\Delta n)^2$5 (Khwaja et al., 2016).

3. Quadratic transmittance as quadratic phase in paraxial optics

A different usage appears in paraxial propagation theory, where quadratic transmittance means a multiplicative quadratic factor in transverse coordinates. In the unified paraxial formulation for free space, linear-index media, and quadratic-index media, the quadratic-index case obeys

$(\Delta n)^2$6

while free space corresponds to $(\Delta n)^2$7. The quadratic medium is thus mathematically identical to a Schrödinger equation with a harmonic-oscillator potential (Korneev et al., 2024).

The Lie-algebraic description uses

$(\Delta n)^2$8

with commutators

$(\Delta n)^2$9

These generate an $O(\sqrt{\eta})$0 algebra, and the propagation operator can be factorized as

$O(\sqrt{\eta})$1

In this decomposition, $O(\sqrt{\eta})$2 is the quadratic transmittance factor, while $O(\sqrt{\eta})$3 is free-space-like propagation and $O(\sqrt{\eta})$4 is a squeezing operator (Korneev et al., 2024).

For free-space propagation of an initial field with quadratic modulation,

$O(\sqrt{\eta})$5

the factor $O(\sqrt{\eta})$6 is explicitly identified as a quadratic transmittance applied at $O(\sqrt{\eta})$7. The factorized solution is

$O(\sqrt{\eta})$8

If $O(\sqrt{\eta})$9, the exponential becomes a pure quadratic phase, equivalent to the thin-lens transmittance

$O(\eta)$0

This identifies quadratic transmittance with the standard lens-like phase element of Fourier optics (Korneev et al., 2024).

The same structure emerges for a quadratic-index, or GRIN, medium. With

$O(\eta)$1

factorization yields

$O(\eta)$2

Hence a finite GRIN segment is equivalent to a sequence of scaling, free-space-like propagation, and a quadratic phase factor

$O(\eta)$3

The corresponding mapping from a free-space solution $O(\eta)$4 to a GRIN solution is

$O(\eta)$5

This provides an exact paraxial equivalence between propagation in a quadratic refractive-index distribution and a generalized quadratic phase element within an SL(2,$O(\eta)$6) or ABCD-like formalism (Korneev et al., 2024).

4. Quadratic dispersion and anisotropic optical transmittance in 2D materials

In anisotropic graphene-like materials and black phosphorus-related systems, quadratic transmittance arises from a semi-Dirac spectrum that is parabolic in one direction and linear in the orthogonal direction. Near the merging of two Dirac nodes, the effective low-energy Hamiltonian is

$O(\eta)$7

with eigenvalues

$O(\eta)$8

Along $O(\eta)$9, the dispersion is parabolic,

$T(d)=T(-d)$0

whereas along $T(d)=T(-d)$1 it is linear,

$T(d)=T(-d)$2

This linear–quadratic node is generated by uniaxial strain that moves and merges Dirac points (Nualpijit et al., 2017).

The anisotropic optical conductivity follows from the Kubo formula. In the low-frequency regime,

$T(d)=T(-d)$3

with

$T(d)=T(-d)$4

Consequently,

$T(d)=T(-d)$5

The quadratic, or parabolic, electronic direction therefore becomes effectively transparent at low frequency, while the linear direction becomes highly reflective (Nualpijit et al., 2017).

For normal incidence on the 2D sheet, the transmittance for polarization angle $T(d)=T(-d)$6 is

$T(d)=T(-d)$7

with

$T(d)=T(-d)$8

where $T(d)=T(-d)$9. As $d^2$0,

$d^2$1

The sheet therefore becomes nearly perfectly transparent for polarization along the quadratic axis and nearly opaque for polarization along the linear axis. The paper states that the transmittance of linearly polarized incident light varies from nearly $d^2$2 to almost $d^2$3 in the microwave and far-infrared regime (Nualpijit et al., 2017).

This behavior differs qualitatively from isotropic graphene, whose optical conductivity is essentially frequency-independent over a broad range and whose transmittance is nearly polarization-independent. In the semi-Dirac system, the quadratic direction does not make $d^2$4 a quadratic polynomial in frequency; rather, it imposes a band-structure anisotropy that drives $d^2$5 conductivity scaling and extreme polarization selectivity (Nualpijit et al., 2017).

5. Quadratic loss mechanisms in transparent piezoelectric ceramics

In transparent piezoelectric ceramics, transmittance is modeled as

$d^2$6

or equivalently

$d^2$7

where $d^2$8 is total reflection loss, $d^2$9 is the total scattering coefficient, $\exp(b_2 J_2)$0 is the absorption coefficient, and $\exp(b_2 J_2)$1 is sample thickness. The problem is motivated by a stated controversy over the influence of grains and pores on ceramic transmittance and by arbitrariness in estimating the theoretical transmittance limit (Xiong et al., 30 Apr 2025).

Xiong et al. refine the reflection limit by introducing an inhomogeneous reflection model in which the ceramic is treated as a stack of grains with different refractive indices along the propagation direction. At each grain boundary, reflection depends on the local mismatch; for small mismatch,

$\exp(b_2 J_2)$2

Thus the local reflection intensity is quadratic in refractive-index mismatch. The global reflection $\exp(b_2 J_2)$3 is a more complicated function of the set of interface reflection and transmission coefficients, but the data indicate that for small mismatch its dependence on birefringence is effectively quadratic at each interface (Xiong et al., 30 Apr 2025).

Scattering is modeled through

$\exp(b_2 J_2)$4

with volume fraction $\exp(b_2 J_2)$5, scatterer diameter $\exp(b_2 J_2)$6, and scattering efficiency $\exp(b_2 J_2)$7. For pores, the Rayleigh approximation gives

$\exp(b_2 J_2)$8

so that

$\exp(b_2 J_2)$9

The contrast factor is approximately quadratic in $O(\eta)$00. For grains, the Rayleigh–Gans–Debye approximation gives

$O(\eta)$01

leading to

$O(\eta)$02

and, explicitly in terms of birefringence,

$O(\eta)$03

This is the clearest quadratic dependence in the ceramic problem: attenuation from grains is proportional to $O(\eta)$04 (Xiong et al., 30 Apr 2025).

The total scattering coefficient is assumed additive,

$O(\eta)$05

and the combined model is summarized as

$O(\eta)$06

In the weak-scattering limit, expanding the exponential yields

$O(\eta)$07

so quadratic corrections also appear in the composite scattering strength itself. The reported fits to PMN-PT data gave $O(\eta)$08 and $O(\eta)$09, implying $O(\eta)$10 for measured average grain size $O(\eta)$11 (Xiong et al., 30 Apr 2025).

In this domain, “quadratic transmittance” is therefore best understood as a shorthand for quadratic loss channels rather than a standalone transmission law: grain-boundary reflection and grain scattering are both governed by $O(\eta)$12, while pore scattering adds distinct $O(\eta)$13 and $O(\eta)$14 scaling (Xiong et al., 30 Apr 2025).

6. Quadratic improvement in rate–transmittance scaling for QKD

In long-distance quantum key distribution, the relevant transmittance is the overall probability $O(\eta)$15 that a photon is successfully transmitted through the channel and detected. Conventional point-to-point and standard MDI-QKD were long understood to be limited by a linear rate-transmittance relation,

$O(\eta)$16

with the PLOB bound

$O(\eta)$17

In symmetric MDI-QKD, each arm has transmittance $O(\eta)$18, and useful events require a photon from each side, so the detection rate scales as $O(\eta)$19 (Fang et al., 2019).

Phase-matching QKD alters this scaling. Alice and Bob encode information in the phase of coherent states on a single optical mode, and the middle node detects only the relative phase. The critical statement is that Charlie needs only one photon in the joint quantum system, rather than one from each side simultaneously. The resulting secure key rate obeys

$O(\eta)$20

which the paper describes as a quadratic improvement over original MDI-QKD. In this setting, “quadratic” does not mean a quadratic polynomial in $O(\eta)$21; it refers to the exponent improvement from $O(\eta)$22 to $O(\eta)$23 in the rate-transmittance law (Fang et al., 2019).

The implemented PM-QKD protocol uses phase slicing, weak coherent signal and decoy states, an untrusted middle measurement node, laser injection locking, and phase post-compensation. The key-rate formula for signal states is

$O(\eta)$24

with

$O(\eta)$25

and the per-pulse key rate is $O(\eta)$26. Experimentally, the system surpassed the linear PLOB bound over 302 km and 402 km commercial-fibre channels and yielded a secret key rate of $O(\eta)$27 bps over 502 km ultralow-loss fibre (Fang et al., 2019).

The reported transmittances and bound comparisons are explicit. At 302 km, the effective total transmittance was $O(\eta)$28, the PLOB bound was $O(\eta)$29, and the experimental asymptotic key rate was $O(\eta)$30, which was stated to be $O(\eta)$31 above the bound. At 502 km, the effective total transmittance was $O(\eta)$32 and the experiment still produced a positive key rate. The paper further notes that this does not violate the secret-key capacity of a single pure-loss channel, because PM-QKD uses a two-leg network geometry with an intermediate measurement node rather than a single direct channel (Fang et al., 2019).

Taken together, these examples show that quadratic transmittance is not a monolithic concept but a recurrent structural motif. Depending on context, it may denote a quadratic misalignment loss, a quadratic phase operator, an optical response set by a quadratic band curvature, attenuation quadratic in refractive-index mismatch, or a square-root transmission law that constitutes a quadratic improvement over linear scaling.