Quantized Paraxial Framework

• Quantized Paraxial Framework is an approach that unifies paraxial optics with quantum translation operators to rigorously describe beam evolution in optical systems.
• It extends the traditional ABCD matrix formalism to include higher-order corrections, enabling precise modeling of complex and nonlinear beam trajectories.
• The framework bridges classical optics and quantum mechanics, facilitating quantum state propagation and the systematic inclusion of aberrations in optical design.

A quantized paraxial framework refers to the theoretical and mathematical structures that unify paraxial optics—the regime of small angular deviations from a principal axis—with quantum mechanics, particularly through the formal identification of optical propagation operators with quantum translation and evolution operators. This approach not only extends the classic ABCD matrix formalism used in ray optics to arbitrary beam trajectories and higher-order corrections, but also establishes a one-to-one correspondence between paraxial optical evolution and fundamental quantum-mechanical propagators.

1. ABCD Matrix Formalism and Paraxial Propagation Operators

In canonical paraxial optics, the evolution of a light beam or ray through free space or linear optical systems is described by the ABCD matrix formalism. A light ray with position and angle parameters is represented by a two-component vector,

$$\mathbf{v} = \begin{pmatrix} f(x) \ f'(x) \end{pmatrix}$$

with propagation from $x_1$ to $x_2 = x_1 + L$ governed by the matrix equation: $$\begin{pmatrix} f(x_2) \ f'(x_2) \end{pmatrix} = \begin{bmatrix} 1 & L \ 0 & 1 \end{bmatrix} \begin{pmatrix} f(x_1) \ f'(x_1) \end{pmatrix}$$ This form robustly expresses the first-order (linear) phase space transformation of beam parameters in the paraxial regime, where the small-angle approximation holds.

2. Generalized Matrix Formalism for Arbitrary Trajectories

The framework introduced in "Classical optics representation of the quantum mechanical translation operator via ABCD matrices" (Ornigotti et al., 2012) significantly generalizes the ABCD approach to handle higher-order (nonlinear) beam evolution:

• The Taylor expansion of the ray trajectory is carried to arbitrary order: $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots$.
• By re-expanding about different transverse planes ($x_i$, $x_j$), one derives a linear transformation between the expansion coefficients, represented succinctly as:

$B(x_j) \mathbf{a}(x_j) = B(x_i) \mathbf{a}(x_i)$

where $B(x)$ collects the Taylor monomials at $x$ and $\mathbf{a}(x)$ is the vector of Taylor coefficients at $x$.

• This leads to the generalized ABCD matrix:

$A(L) = B^{-1}(x_j) B(x_i)$

which in the principal 2×2 block coincides with the ordinary ABCD matrix, but in full contains all orders—effectively capturing the nonlinearity of inhomogeneous media or complex trajectory evolution.

Notably, the matrix $A(L)$ admits a power-series expansion: $A(L) = \sum_{k=0}^\infty \frac{(L D)^k}{k!}$ where $D$ is the infinite-dimensional matrix representation of the differentiation operator, corresponding to $d/dx$.

3. Correspondence with the Quantum Mechanical Translation Operator

The core theoretical advance is the identification: $A(L) = e^{L D}$ which is mathematically identical to the quantum translation operator

$T(L) = e^{L \frac{d}{dx}}$

familiar from quantum mechanics, where it acts on a state $f(x)$ to yield $f(x+L)$. Thus, the expanded paraxial operator is isomorphic to translation in the Hilbert space of wavefunctions. The semigroup property

$A(L_1) A(L_2) = A(L_1 + L_2)$

further confirms the group-theoretic parallel with quantum translations.

This result establishes a rigorous optical analog of the quantum propagator for state evolution under translation—uniting beam propagation in classical optics and quantum kinematics in a single algebraic structure.

4. Implications for Quantization and Quantum Optics

The quantized paraxial framework extends naturally to quantum optics:

• The analogy invites direct application of quantum-mechanical tools (e.g., operator formalism, Hilbert space representations) to paraxial field evolution.
• It suggests quantization of the paraxial wave equation can proceed by promoting the generalized beams' mode coefficients to operators, with propagation governed by the translation operator $e^{L D}$.
• This relation underpins quantum state evolution in paraxial optical systems, and enables systematic inclusion of higher-order (aberration, inhomogeneity) effects in quantum field theories of light.
• For quantum information tasks and advanced optical measurements, the unification provides a route to describe the quantum evolution of field and state under arbitrary paraxial propagation.

5. Formulas and Rigorous Statements

Concept	Mathematical Expression	Context/Reference
Standard ABCD propagation (lowest order)	$$\begin{pmatrix} f(x_2) \ f'(x_2) \end{pmatrix} = \begin{bmatrix} 1 & L \ 0 & 1 \end{bmatrix} \begin{pmatrix} f(x_1) \ f'(x_1) \end{pmatrix}$$	Eq. (2), (Ornigotti et al., 2012)
Weighting of Taylor coefficients	$B(x_j) \mathbf{a}(x_j) = B(x_i) \mathbf{a}(x_i)$	Eq. (17), (Ornigotti et al., 2012)
Generalized propagation matrix	$A(L) = B^{-1}(x_j) B(x_i)$	Eq. (19), (Ornigotti et al., 2012)
Series expansion as exponential operator	$$A(L) = \sum_{k=0}^{\infty} \frac{(L D)^k}{k!} = e^{L D}$$	Eq. (25), (Ornigotti et al., 2012)
Quantum translation operator	$T(L) = e^{L \frac{d}{dx}}$	Section 3, (Ornigotti et al., 2012)
Semigroup property	$A(L_1) A(L_2) = A(L_1 + L_2)$	Section 3, (Ornigotti et al., 2012)

6. Practical Applications

• Precision Optical Design: The extended formalism systematically incorporates aberrations, higher-order geometrical effects, and propagation through inhomogeneous media, providing direct computational tools for rigorous lens and beamline models.
• Quantum State Propagation: By establishing the generalized ABCD matrix as the translation operator, optical systems can be mapped onto quantum propagators. This is invaluable for quantum information processing and for developing quantum light sources with engineered propagation characteristics.
• Simulation and Modeling: The algebraic framework is amenable to both analytical and numerical implementation; operator exponentiation mirrors quantum approaches, facilitating transfer of computational techniques between optics and quantum mechanics.

7. Theoretical and Foundational Significance

The quantized paraxial framework rigorously links ray and wave optics with quantum mechanics. This correspondence does not merely provide an analogy but delivers a physically and mathematically exact bridge: the full expansion of beam trajectories corresponds to the action of quantum-mechanical translation, with the paraxial evolution operator $e^{L D}$ playing the dual role of classical coordinate transformation and quantum propagator.

This insight underwrites the translation of methods and intuitions between classical beam physics and quantum state evolution, enabling unified developments in both classical and quantum regimes of optical and wave science.