Divergence Amplification Factor

• Divergence Amplification Factor is a measure capturing how a system’s response diverges or saturates as a control parameter nears a critical or singular boundary.
• In optics it quantifies field amplitude ratios at critical incidence, while in non-Hermitian systems it describes geometric intensity changes and shifts in Petermann factors.
• In differential privacy it frames a worst-case Rényi-divergence amplification via Bernoulli post-sampling, guiding regularization and mechanism optimization.

Searching arXiv for the cited papers to ground the article in current records. Divergence amplification factor denotes a quantity that tracks how a measurable response grows, saturates, or is effectively transformed when an underlying control parameter approaches a critical, singular, or information-theoretic boundary. In the cited literature, the term appears in three distinct technical settings: the field-amplitude amplification factor $F_{\rm amp}(\theta)$ for light modes in a homogeneous interface layer near critical conditions (Sigel, 15 Jun 2026), the geometric amplification factor $A_{\rm geo}[{\cal C}]$ in non-Hermitian adiabatic evolution (Ozawa et al., 2024), and the worst-case amplification curve $Post_{d,k,\alpha}(\epsilon)$ governing Rényi-divergence behavior under Bernoulli post-sampling in differential privacy (Imola et al., 2021). The shared vocabulary reflects the presence of a divergence or near-divergence structure, but the underlying objects, observables, and operational meanings are different.

1. Domain-specific definitions

The principal definitions are structurally similar in that each introduces a normalized or worst-case factor, but they operate on different mathematical objects.

Setting	Quantity	Definition
Interface optics	$F_{\rm amp}(\theta)$	$F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$
Non-Hermitian adiabatic evolution	$A_{\rm geo}[{\cal C}]$	$$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$
Bernoulli post-sampling and RDP	$Post_{d,k,\alpha}(\epsilon)$	$$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$

In the optical setting, the quantity is a field-amplitude ratio normalized to a reference angle away from criticality. In the non-Hermitian setting, it is the purely geometric contribution to norm change under adiabatic transport. In the privacy setting, it is a worst-case curve defined through a supremum over mechanisms or, equivalently, over distributions $P,Q$ on $A_{\rm geo}[{\cal C}]$0 subject to two Rényi-divergence constraints (Sigel, 15 Jun 2026, Ozawa et al., 2024, Imola et al., 2021).

A common source of ambiguity is that “amplification” does not always mean literal increase of an experimentally measured amplitude. In optics it refers to the divergence of plane-wave basis amplitudes near $A_{\rm geo}[{\cal C}]$1; in non-Hermitian dynamics it refers to norm change induced by the imaginary part of a Berry-type connection; in privacy it measures how much smaller the $A_{\rm geo}[{\cal C}]$2-RDP parameter becomes after Bernoulli post-processing. This suggests that the phrase is best understood as a family resemblance rather than a single invariant notion.

2. Critical-incidence amplification in a homogeneous interface layer

For a homogeneous layer 1 in total reflection geometry, the field-amplitude amplification factor is defined by

$A_{\rm geo}[{\cal C}]$3

where $A_{\rm geo}[{\cal C}]$4 is taken at a reference angle well away from criticality. The normal wave-vector component in layer 1 is

$A_{\rm geo}[{\cal C}]$5

and the amplification behaves as

$A_{\rm geo}[{\cal C}]$6

up to a non-singular prefactor (Sigel, 15 Jun 2026).

The transfer-matrix solution gives the two field coefficients in layer 1 in the schematic form

$A_{\rm geo}[{\cal C}]$7

with common denominator

$A_{\rm geo}[{\cal C}]$8

Expanding numerator and denominator near criticality $A_{\rm geo}[{\cal C}]$9 yields

$Post_{d,k,\alpha}(\epsilon)$0

where $Post_{d,k,\alpha}(\epsilon)$1 is finite and depends on $Post_{d,k,\alpha}(\epsilon)$2 but not on $Post_{d,k,\alpha}(\epsilon)$3. The singular contribution is therefore $Post_{d,k,\alpha}(\epsilon)$4 (Sigel, 15 Jun 2026).

The relevant critical angle in the absence of absorption is

$Post_{d,k,\alpha}(\epsilon)$5

Writing $Post_{d,k,\alpha}(\epsilon)$6, the small-$Post_{d,k,\alpha}(\epsilon)$7 expansion gives

$Post_{d,k,\alpha}(\epsilon)$8

hence

$Post_{d,k,\alpha}(\epsilon)$9

It follows that

$F_{\rm amp}(\theta)$0

The critical exponent is therefore $F_{\rm amp}(\theta)$1 (Sigel, 15 Jun 2026).

The physical picture given for this divergence is that, exactly at critical incidence, the refracted wave in layer 1 travels tangent to the interfaces and “piles up” over an infinite distance, producing a $F_{\rm amp}(\theta)$2 amplification in the plane-wave basis amplitudes. Absorption and finite spatial coherence then act as cutoffs rather than as changes to the singular mechanism itself.

3. Absorption regularization and observable peak structure

In a weakly absorbing layer, one writes $F_{\rm amp}(\theta)$3. The exact $F_{\rm amp}(\theta)$4 then satisfies

$F_{\rm amp}(\theta)$5

The minimum possible $F_{\rm amp}(\theta)$6 occurs at the “absorbing” critical angle

$F_{\rm amp}(\theta)$7

and has value

$F_{\rm amp}(\theta)$8

The divergence is therefore regularized into a finite maximum whose saturated field peak-height is

$F_{\rm amp}(\theta)$9

For the half-width at half-maximum in angle, solving $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$0 gives

$F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$1

identified as the half-width at half maximum of the intensity (Sigel, 15 Jun 2026).

The no-absorption treatment also introduces a peak-base width,

$F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$2

where $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$3 is the evanescent penetration depth in layer 2. This separates two related but distinct notions of width: a base width in the idealized singular setting, and an HWHM once absorption regularizes the singularity (Sigel, 15 Jun 2026).

The experimental significance emphasized for visible-light Evanescent-Wave Dynamic Light Scattering is that the scattered intensity from an interface layer is proportional to $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$4. Without absorption it would diverge as $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$5; with finite absorption and finite beam-profile width, the result is a sharply peaked but finite scattering intensity at $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$6. The paper summary identifies this as the origin of the dramatic $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$7–$F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$8 amplification observed experimentally, and it links the effect to depth-selective (“tomographic”) measurements of interfacial fluctuations. The same summary notes a relation to surface plasmon resonance and states that in SPR second-harmonic generation the SHG field is proportional to $F_{\rm amp}(\theta)\equiv |A(\theta)|/|A_0|$9, so the SHG intensity scales as $A_{\rm geo}[{\cal C}]$0, producing $A_{\rm geo}[{\cal C}]$1 wings in the no-absorption regime (Sigel, 15 Jun 2026).

4. Geometric amplification in non-Hermitian adiabatic evolution

In non-Hermitian adiabatic dynamics, the relevant object is not a spatially diverging field coefficient but a geometric contribution to intensity change along a path in parameter space. Let $A_{\rm geo}[{\cal C}]$2 be a non-Hermitian Hamiltonian with nondegenerate right eigenstate $A_{\rm geo}[{\cal C}]$3 and left eigenstate $A_{\rm geo}[{\cal C}]$4,

$A_{\rm geo}[{\cal C}]$5

Two Berry connections are introduced,

$A_{\rm geo}[{\cal C}]$6

and the non-Hermitian Berry connection is defined by

$A_{\rm geo}[{\cal C}]$7

For adiabatic variation $A_{\rm geo}[{\cal C}]$8, the total intensity amplification factor

$A_{\rm geo}[{\cal C}]$9

splits into a dynamical part, coming from $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$0, and a purely geometric part. The geometric amplification factor is

$$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$1

for a path $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$2 (Ozawa et al., 2024).

Path-independence is controlled by the Berry curvature two-form

$$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$3

If the parameter space is simply-connected, then

$$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$4

is equivalent to the line integral being path-independent, so that $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$5 depends only on the endpoints $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$6 and not on the detailed shape of $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$7 (Ozawa et al., 2024).

This formulation makes the amplification factor a geometric quantity in a precise sense: it is generated by the imaginary part of a Berry-type connection rather than by the instantaneous imaginary part of the eigenvalue. A plausible implication is that “divergence” in this context refers to the accumulation of geometric norm change under adiabatic steering, not to a local singularity in real space.

5. Symmetry classes and Petermann-factor formulations

The non-Hermitian analysis identifies four general situations in which $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$8, often described as the existence of a metric or particle–hole symmetry relating left and right eigenvectors. With $$\exp\!\Bigl[-2\int_{\cal C}\Im[{\cal A}_n(R)]\!\cdot dR\Bigr]$$9 a fixed parameter-independent matrix, these are:

$Post_{d,k,\alpha}(\epsilon)$0

$Post_{d,k,\alpha}(\epsilon)$1

and the inverse relations

$Post_{d,k,\alpha}(\epsilon)$2

$Post_{d,k,\alpha}(\epsilon)$3

In each case,

$Post_{d,k,\alpha}(\epsilon)$4

vanishes identically, and hence $Post_{d,k,\alpha}(\epsilon)$5. The geometric amplification factor is then endpoint-only (Ozawa et al., 2024).

The Petermann factor is

$Post_{d,k,\alpha}(\epsilon)$6

with

$Post_{d,k,\alpha}(\epsilon)$7

When $Post_{d,k,\alpha}(\epsilon)$8 is a rank-1 projector, one obtains the linear ratio

$Post_{d,k,\alpha}(\epsilon)$9

When $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$0 is unitary, and Hermitian or symmetric, one obtains the square-root ratio

$$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$1

The projector case is highlighted because it permits direct measurement of the change in Petermann factor from initial to final parameter (Ozawa et al., 2024).

Two examples are given. For the two-level Hamiltonian

$$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$2

in the real-eigenvalue regime $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$3, the Petermann factor is

$$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$4

When $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$5, the Hamiltonian is unitarily equivalent to a symmetric one, so $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$6 is unitary and $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$7; the same $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$8 is reached for different driving paths in $$\sup_{A:\epsilon_A(\alpha)\le\epsilon}\epsilon_{B_k\circ A}(\alpha)$$9. In the nonreciprocal robotic metamaterial example, an effective non-Hermitian SSH chain has a single zero-mode with right eigenvector $P,Q$0 and left eigenvector chosen fixed by setting $P,Q$1 constant. Then one is in the rank-1 projector case and

$P,Q$2

The same study states that a slow adiabatic experiment can directly extract the final Petermann factor through

$P,Q$3

In a reciprocal realization $P,Q$4, one instead has $P,Q$5 and $P,Q$6 (Ozawa et al., 2024).

6. Divergence amplification in Bernoulli post-sampling and Rényi differential privacy

In the privacy setting, the divergence-amplification factor is formulated as a worst-case curve rather than as a local singular law. Fix $P,Q$7, let $P,Q$8 be any randomized mechanism satisfying $P,Q$9-RDP, and assume $A_{\rm geo}[{\cal C}]$00 outputs $A_{\rm geo}[{\cal C}]$01. After running $A_{\rm geo}[{\cal C}]$02, one draws $A_{\rm geo}[{\cal C}]$03 independent Bernoulli samples with biases given by the coordinates of $A_{\rm geo}[{\cal C}]$04. The Bernoulli-sampling process $A_{\rm geo}[{\cal C}]$05 returns $A_{\rm geo}[{\cal C}]$06 i.i.d. draws $A_{\rm geo}[{\cal C}]$07 with

$A_{\rm geo}[{\cal C}]$08

independently for all $A_{\rm geo}[{\cal C}]$09. Writing

$A_{\rm geo}[{\cal C}]$10

the post-processed mechanism $A_{\rm geo}[{\cal C}]$11 has

$A_{\rm geo}[{\cal C}]$12

The worst-case amplification curve is then

$A_{\rm geo}[{\cal C}]$13

This is the quantity called the divergence-amplification factor in the Bernoulli post-sampling analysis (Imola et al., 2021).

The exact computation begins from an infinite-dimensional optimization over $A_{\rm geo}[{\cal C}]$14. Two reductions are stated. First, by a convexity and “corner-point” argument, worst-case $A_{\rm geo}[{\cal C}]$15 can be taken on

$A_{\rm geo}[{\cal C}]$16

Second, writing the Rényi-divergence constraints and the post-sampling divergence in closed form yields a convex program in the $A_{\rm geo}[{\cal C}]$17 variables $A_{\rm geo}[{\cal C}]$18, where

$A_{\rm geo}[{\cal C}]$19

and

$A_{\rm geo}[{\cal C}]$20

For $A_{\rm geo}[{\cal C}]$21, the constraints are

$A_{\rm geo}[{\cal C}]$22

and for each $A_{\rm geo}[{\cal C}]$23,

$A_{\rm geo}[{\cal C}]$24

with objective

$A_{\rm geo}[{\cal C}]$25

The output is $A_{\rm geo}[{\cal C}]$26. For general $A_{\rm geo}[{\cal C}]$27, $A_{\rm geo}[{\cal C}]$28 is replaced by $A_{\rm geo}[{\cal C}]$29-tuples and the exponents become products over $A_{\rm geo}[{\cal C}]$30. The resulting program has $A_{\rm geo}[{\cal C}]$31 variables and is solvable for small $A_{\rm geo}[{\cal C}]$32 by standard convex solvers (Imola et al., 2021).

The upper and lower bounds given for the amplification curve are

$A_{\rm geo}[{\cal C}]$33

where

$A_{\rm geo}[{\cal C}]$34

The $A_{\rm geo}[{\cal C}]$35 term follows from post-processing, while the $A_{\rm geo}[{\cal C}]$36 term comes from quasi-convexity of $A_{\rm geo}[{\cal C}]$37 and independence across coordinates. The lower bound uses a two-point construction placing mass $A_{\rm geo}[{\cal C}]$38 and $A_{\rm geo}[{\cal C}]$39 on the all-$A_{\rm geo}[{\cal C}]$40 and all-$A_{\rm geo}[{\cal C}]$41 corners, which gives $A_{\rm geo}[{\cal C}]$42 and

$A_{\rm geo}[{\cal C}]$43

with

$A_{\rm geo}[{\cal C}]$44

$A_{\rm geo}[{\cal C}]$45

The summary states that in many regimes, including large $A_{\rm geo}[{\cal C}]$46 or moderate-to-large $A_{\rm geo}[{\cal C}]$47, the upper and lower bounds coincide or become arbitrarily close, yielding either $A_{\rm geo}[{\cal C}]$48 or $A_{\rm geo}[{\cal C}]$49 depending on parameters (Imola et al., 2021).

A worked example is provided for $A_{\rm geo}[{\cal C}]$50, $A_{\rm geo}[{\cal C}]$51, $A_{\rm geo}[{\cal C}]$52, $A_{\rm geo}[{\cal C}]$53, $A_{\rm geo}[{\cal C}]$54. In that case,

$A_{\rm geo}[{\cal C}]$55

the upper bound gives

$A_{\rm geo}[{\cal C}]$56

the lower bound yields

$A_{\rm geo}[{\cal C}]$57

and the convex solver finds $A_{\rm geo}[{\cal C}]$58, so the two-point lower bound is tight in that case (Imola et al., 2021).

7. Comparative interpretation and limits of unification

Across these settings, the divergence amplification factor always quantifies a transformed response under a limiting process, but the limiting processes are not the same. In interface optics, the relevant limit is $A_{\rm geo}[{\cal C}]$59 at critical incidence, giving $A_{\rm geo}[{\cal C}]$60 and $A_{\rm geo}[{\cal C}]$61 in the absence of absorption (Sigel, 15 Jun 2026). In non-Hermitian adiabatic transport, the factor is a path integral of $A_{\rm geo}[{\cal C}]$62, with path-independence controlled by $A_{\rm geo}[{\cal C}]$63 and, in specific symmetry classes, reducible to ratios of Petermann factors (Ozawa et al., 2024). In Bernoulli post-sampling, the factor is a worst-case Rényi-divergence curve obtained from a supremum over admissible mechanisms or distributions, with exact convex-program computation and upper/lower bounds (Imola et al., 2021).

A frequent misconception would be to treat these as interchangeable manifestations of a single formalism. The available material does not support that. The optical quantity is a normalized field amplitude in a transfer-matrix treatment; the non-Hermitian quantity is an adiabatic geometric norm factor; the privacy quantity is an extremal divergence functional over randomized mechanisms. What unifies them is not a common algebraic object but the repeated appearance of singular, endpoint, or worst-case amplification behavior under precisely defined constraints.

A plausible implication is that the phrase “divergence amplification factor” is best reserved for context-specific use, with the relevant state space, observable, and regularization mechanism made explicit. In the optical case, absorption and finite spatial coherence cut off the divergence to a finite peak. In the non-Hermitian case, the central distinction is between dynamical amplification from $A_{\rm geo}[{\cal C}]$64 and geometric amplification from $A_{\rm geo}[{\cal C}]$65. In the privacy case, the central distinction is between exact worst-case optimization and computable bounds such as $A_{\rm geo}[{\cal C}]$66. These distinctions determine both the interpretation and the practical use of the factor in each domain.