Closed-Form Optimal Deflection Angle

• Closed-form optimal deflection angle is an analytic formula expressed solely in terms of measurable geometric or physical parameters to predict ray deviations.
• It unifies weak and strong field regimes using asymptotic approximants and series expansions, eliminating the need for iterative numerical methods.
• This approach enhances precision and interpretability in applications like gravitational lensing, astrometry, and optical beam steering.

A closed-form optimal deflection angle is a comprehensive analytic formula for the angular deviation of rays (light, particles, or beams) due to a refracting, gravitational, or geometric interface. Such formulas are designed to be maximally accurate within their domain—encompassing both weak and strong field or perturbative and non-perturbative regimes—and are expressed purely in terms of algebraic functions, elementary transcendental functions, or standard special functions of geometric or physical parameters. These formulas play a central role in both experimental and theoretical analyses where iterative integration, root finding, or computational inversion are prohibitive or unnecessary, and offer superior scalability and interpretability for high-precision modeling in, e.g., lensing, beam optics, astrometry, and wave propagation.

1. General Principles of Closed-Form Deflection Angles

Closed-form optimal deflection angles exist across domains and arise from first integrals of the governing equations of motion—either geodesic, optical, or wave equations. The essential features include:

• Expression strictly in terms of physically measurable quantities (such as impact parameter $b$, object radius $a$, or distance parameters).
• Uniform accuracy across multiple regimes (weak/strong field, paraxial/non-paraxial optics, finite/infinite source or observer distances).
• Avoidance of iterative numerical procedures (integrals, roots, quadrature), favoring explicit relationships and, where necessary, series expansions with controlled remainder.
• Manifest covariance and parameter robustness (no coordinate ambiguities or artificial singularities) (Bertone et al., 2011, Nakajima et al., 2012, Tsukamoto, 2016, Barlow et al., 2017, Igata, 3 May 2025).

These formulas are derived either directly from the underlying physics (metric, refractive index profile) or via higher-level methods such as asymptotic matching, time-transfer functions, or auxiliary-function algorithms.

2. Closed-Form Deflection in Schwarzschild and Wormhole Geometries

For null geodesics in an Ellis wormhole, the metric yields a closed form for the deflection angle via elliptic integrals:

$\alpha(b) = 2 K(a / b) - \pi$

where $K(k)$ is the complete elliptic integral of the first kind, $a$ is the throat radius, and $b \ge a$ is the impact parameter. Weak-field expansions yield

$$\alpha(b) = \frac{\pi}{4} \left(\frac{a}{b}\right)^2 + \frac{9\pi}{64}\left(\frac{a}{b}\right)^4 + O\left(\frac{a^6}{b^6}\right)$$

valid for $a/b \lesssim 0.3–0.5$, while strong deflection ($b \to a$) produces a logarithmic divergence, reflecting the topology of the wormhole (Nakajima et al., 2012). For Schwarzschild and Reissner–Nordström black holes, the strong-deflection limit (SDL) takes the universal form (Tsukamoto, 2016):

$$\alpha(b) = -\bar a \ln\left(\frac{b}{b_c} - 1\right) + \bar b + O\left((b-b_c)\ln(b-b_c)\right)$$

with explicit formulas for $\bar a$, $\bar b$ given in terms of metric derivatives at the photon sphere. This analytic structure is essential for lensing near compact objects.

3. Optimality via Asymptotic Approximants in Kerr Spacetimes

For equatorial photon trajectories in Kerr black holes, optimal closed-form deflection angles are constructed through asymptotic approximants matching both weak-field and strong-field behavior. The general form is (Barlow et al., 2017, Beachley et al., 2018):

$$\alpha_A^N(b, a) = -\pi + \beta(a) + \gamma(a) \ln \zeta(a) + \delta_{a, 1}[\sqrt{3}/b' - \gamma(a)\ln b'] + \sum_{n=1}^{N+1} B_n(a) b'^{n/2}\left[\Delta_{n+1}\sqrt{b'}\ln b' + \Delta_n\right]$$

where $b' = 1 - b_c(a)/b$, $N$ is matching order, and the coefficients $\beta(a), \gamma(a), \zeta(a), B_n(a)$ are determined to enforce strong-deflection divergence and weak-field Taylor expansions precisely. For $N=4$, the relative error is $\lesssim 2\times10^{-4}$ over the full $b′$ domain except for extremal spin, where it is $\sim10^{-6}$ (Barlow et al., 2017). The method eliminates elliptic integrals and is computationally optimal for ray-tracing.

4. Coordinate-Invariant SDL Coefficient and ZAMO Formulation

Recent advances demonstrate that the key logarithmic-divergence coefficient in the strong deflection limit, denoted $\bar a$, can be expressed entirely in terms of local invariant quantities measurable by zero-angular-momentum observers (ZAMO). Specifically, for stationary axisymmetric spacetimes—including Kerr and generic rotating lenses—the critical parameter is (Igata, 3 May 2025):

$\bar a_{\pm} = \sqrt{-\frac{2}{V''_{m,\pm}}}$

where $V''_{m,\pm}$ is the second derivative of the effective potential at the photon sphere and can be written as

$$V''_{m,\pm} = [R_m(1 \pm v_{m,\pm})]^2 \left\{ G_{(0)(0)}^{m,\pm} + G_{(3)(3)}^{m,\pm} - 2(E_{(2)(2)}^{m,\pm} - E_{(1)(1)}^{m,\pm}) \pm [4 B_{(1)(2)}^{m,\pm} + 2 G_{(0)(3)}^{m,\pm}] \right\}$$

with curvature tensors $E$, $B$, $G$ taken in the ZAMO frame at the photon orbit. This form ensures coordinate independence and connects directly to observable physical quantities and the damping rate of quasinormal modes in the eikonal limit.

5. Astrometric and Geometric Optimization: Maximum Deflection

In astrometric applications (finite source/receiver distance, e.g., Gaia, SIM), the closed-form deflection angle for weak, static spherical fields is (Bertone et al., 2011):

$$\Delta\theta = (1+\gamma)\frac{GM}{c^2 b} \left[ \sqrt{1 - \frac{b^2}{r_A^2}} + \sqrt{1 - \frac{b^2}{r_B^2}} \right]$$

where $r_A$, $r_B$ are distances from the mass to emitter/observer. The maximal deflection is achieved for $b \to b_{\min}$ (grazing incidence) and $r_A$, $r_B \gg b$, yielding

$$\Delta\theta_{\text{max}} = 2(1+\gamma)\frac{GM}{c^2 b_{\min}}$$

with $b_{\min}$ typically set by the physical radius of the lensing body. Deviations due to finite distance or non-grazing geometry reduce $\Delta\theta$.

6. Optical Beam Deflection: Universal Closed-Form Formulation

For angular deviations of optical beams (e.g., at Brewster angles in dielectric media), first-order Taylor expansions of Fresnel coefficients yield a universal cubic equation for the peak position. In the regime near Brewster, the depressed cubic equation for the scaled coordinate $\rho$ is (Leo et al., 2021):

$$4\rho^3 + 4\delta\rho^2 + (\delta^2 - 2)\rho = \delta$$

with dimensionless detuning $\delta = (\theta - \theta_B) k w$, independent of specific material properties. The closed-form roots specify the optimal beam deflection position and provide direct laboratory formulas for weak value amplification and nonlinear beam steering.

7. Methods for Weak-Field Closed-Form Deflection

Auxiliary-function methods provide efficient closed-form deflection angles for weak-field (large $b$) geodesics, including charged and timelike orbits in Kerr–Newman spacetimes (Li, 23 Jan 2024):

$$\delta(b) = 2(1+v^{-2})\,\frac{M}{b} - 2\,\frac{qQ}{E v^2}\,\frac{1}{b} + \frac{3\pi}{4b^2}[(1+4v^{-2})M^2 - \frac{q^2 Q^2}{E^2 v^4}] + O(b^{-3})$$

where $v$ is the velocity parameter and $a$ (spin) enters at higher order. This approach eliminates iterative root-finding and provides closed-form accuracy in broad multi-parameter backgrounds.

Table: Representative Closed-Form Deflection Angles

Physical Context	Formula Type	Reference
Ellis wormhole	Elliptic integral	$\alpha(b) = 2 K(a/b) - \pi$ (Nakajima et al., 2012)
Schwarzschild/Kerr SDL	Logarithmic expansion	$\alpha(b) = -\bar a \ln(\dots) + \bar b$ (Tsukamoto, 2016, Igata, 3 May 2025)
Kerr (equatorial, general spin)	Asymptotic algebraic-logarithmic	$\alpha_{A}^{N}(b,a)$ (Barlow et al., 2017, Beachley et al., 2018)
Astrometric weak-field	Algebraic square-root structure	$\Delta\theta = \dots$ (Bertone et al., 2011)
Optical beam deviation	Depressed cubic	$$4\rho^3 + 4\delta\rho^2 + (\delta^2 - 2)\rho = \delta$$ (Leo et al., 2021)
Kerr–Newman weak-field	Algebraic, impact-parameter-dependent	$\delta(b) = \dots$ (Li, 23 Jan 2024)

The closed-form optimal deflection angle encapsulates the precise analytic mapping between geometric, physical, or experimental parameters and observable angular deviation, bridging astrophysical lensing, precision space astrometry, wave optics, and beam steering. It ensures maximal modeling efficiency, accuracy, and interpretability, particularly in regimes inaccessible to purely perturbative or numerical schemes, and underpins contemporary developments in both gravitational theory and experimental photonics.