Simulates a non-linear regression workflow on corrupted experimental data. Diagnoses the divergent effects of random measurement noise, systematic alignment offsets, and uncalibrated geometric parameters on the extracted refractive index.
System Parameters
Target Material
True bulk refractive index. Overridden if a specific wavelength is selected.
Geometric Calibration
True physical angle (Solver assumes exactly 60°).
Table Alignment
Rigidly shifts data (e.g., inaccurate normal reference).
Visual Precision
Gaussian scatter scale multiplier.
Extracted Index (nglass)
--
True Value: --
Extraction Error (Δn)
--
Deviation from True Index
Non-Linear Regression Fit
Fit Residuals (Diagnostics)
The Angle of Deviation Curve-Fitting Diagnostic
1. Forward Kinematics vs. Inverse Regression
A forward kinematic simulation maps a known cause to a theoretical effect (e.g., mapping a known refractive index to a specific continuous curve). The curve-fitting method fundamentally reverses this direction. The solver takes a discrete array of noisy physical measurements and applies an optimization algorithm to extract the unknown cause (the refractive index) by minimizing the sum of squared residuals between the theoretical geometry and the raw scatter data.
2. Metrological Vulnerabilities
Different types of experimental error propagate through optimization algorithms differently:
Random Measurement Noise (σδ): Occurs from the physical difficulty of visually aligning reticles on a diverging laser spot. Because the noise is symmetrically distributed across the entire dataset, the statistical regression heavily suppresses the error. The extracted refractive index remains highly accurate.
Systematic Zero-Offset (θoff): Occurs when the goniometer's 0° baseline is not strictly normal to the prism face. This error rigidly shifts the entire experimental dataset horizontally. The non-linear solver, blind to the mechanical misalignment, is mathematically forced to contort the theoretical curve to match the shifted data. This physical mismatch produces a distinct parabolic "bowing" signature in the residuals.
Uncalibrated Apex Angle (A): The true apex angle of a physical prism rarely matches the nominal specification (e.g., exactly 60°). If the data is generated by a prism with A = 60.1°, but the solver assumes A = 60.0°, the solver will adjust the refractive index to absorb the geometric discrepancy. The fit will appear nearly perfect, and the residuals will remain flat, but the extracted refractive index will be highly inaccurate. This introduces a hidden systematic error that bypasses standard residual diagnostics.
3. Identifying Error Signatures in Residuals
The secondary diagnostic plot tracks the residual error (the vertical distance between each data point and the solver's best-fit curve). By analyzing the shape of these residuals, the experimenter can diagnose the health of the calibration:
Random Scatter: If the residuals are evenly and randomly distributed around the zero axis, it indicates that the analytical model correctly matches the physical geometry. The only active error is visual measurement noise, which the regression successfully averages out.
Parabolic Bowing: If the residuals exhibit a distinct "U-shape" or bowing, it proves the solver is attempting to fit a symmetric mathematical model to asymmetrically shifted physical data. This acts as a visual alert for a systematic zero-offset error (θoff).
Flat Scatter: If an uncalibrated apex angle (A) is present, the residuals will deceptively mimic the ideal state (flat and randomly scattered). The solver successfully forces a fit by manipulating the extracted refractive index to absorb the geometric discrepancy. Because the residuals exhibit no structural anomalies, this demonstrates why physically measuring the prism's apex angle independently is mathematically non-negotiable prior to executing a multi-point regression.