{
  "title": "Paraxial rays in a prescribed linear transverse index",
  "status": "Hypothetical analytical illustration and independent numerical integration",
  "rows": {
    "rays": 603,
    "temperature_index": 61,
    "convergence": 6
  },
  "parameters": {
    "n0": 1.00027,
    "gradient_per_m": [
      -0.002,
      0,
      0.002
    ],
    "distance_m": [
      0,
      2
    ],
    "initial_y_m": 0,
    "initial_slope": 0,
    "temperature_reference_K": 293,
    "index_reference": 1.00027,
    "temperature_range_K": [
      273,
      333
    ]
  },
  "equations": {
    "index": "n(y)=n0+g*y",
    "approximate_ODE": "d\u00b2y/dx\u00b2=g/n0",
    "analytic_reference": "y=y0+slope0*x+g*x\u00b2/(2*n0); dy/dx=slope0+g*x/n0",
    "temperature_index": "n(T)-1=(nref-1)*Tref/T at fixed pressure/composition"
  },
  "checks": {
    "all_pass": true,
    "finite_difference_acceleration_max_abs_error_per_m": 1.0224460167407301e-14,
    "Euler_first_order_error_halving": true,
    "gradient_sign_symmetry": true,
    "zero_gradient_straight_ray": true,
    "index_positive": true,
    "temperature_reference_identity": true,
    "temperature_index_monotone": true
  },
  "endpoint_positive_gradient": {
    "x_m": 2,
    "gradient_per_m": 0.002,
    "y_m": 0.003998920291521289,
    "slope_dy_dx": 0.003998920291521289,
    "index_at_ray": 1.0002779978405831
  },
  "limitations": [
    "The analytic reference exactly solves the stated approximate ODE, not the full eikonal ray equation.",
    "Assumes small slope and small index variation so n in the denominator can be replaced by n0; curvature terms and higher-order corrections are omitted.",
    "The prescribed linear gradient is not a computed temperature, density or flow field. No turbulence, diffraction, boundaries or material dispersion is included.",
    "The optional temperature relation assumes weak refractivity proportional to density, ideal-gas density at fixed pressure/composition, and fixed optical wavelength. Reference index is illustrative, not an air refractivity standard.",
    "Temperature relation does not specify a spatial temperature gradient and is not used to generate the rays.",
    "Solver convergence verifies numerical implementation only; it cannot establish device feasibility or atmospheric accuracy."
  ],
  "applications": [
    "Established context: understanding gradient-index optics and qualitative ray bending in nonuniform media.",
    "Potential research use: a first sensitivity calculation before measured index maps, wave optics and device validation; no improvement or performance is demonstrated."
  ]
}