Polar RMS Error (PE) in Instrument Design

• Polar RMS Error (PE) is a metric that quantifies the root-mean-square deviation in a measured Stokes vector due to uncertainties in optical components.
• It propagates first-order parameter errors through modulation and demodulation matrices to assess instrument accuracy, supporting tolerance allocation.
• Its analytical framework enables precise sensitivity studies and trade-off analysis in polarimetric system design, ensuring compliance with performance requirements.

Polar RMS Error (PE) quantifies the root-mean-square deviation in the Stokes vector measured by a polarimetric system due to the cumulative effect of physical parameter uncertainties in its optical components. This scalar metric provides a comprehensive single-number figure of merit for polarimetric accuracy as propagated through the full modulation–demodulation pipeline, allowing direct comparison with system-level requirements and facilitating tolerance allocation and error budgeting within instrument design (Ovelar et al., 2012).

1. Mathematical Definition of Polar RMS Error

The polarimetric RMS error (PE) is rigorously defined with respect to a reference or “design” input Stokes vector, $S_{\text{in,def}}$. The measured vector, $S_{\text{in}}^*$, is related to the ideal through a total system response

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$

where $\Delta X$ is a $4 \times 4$ matrix collecting all first-order perturbations from imperfect optics and systematics. The error vector is

$$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$

The squared norm is then

$$\|\epsilon\|^2 = \sum_{k=1}^4 [\Delta X S_{\text{in,def}}]_k^2$$

and the polarimetric RMS error is

$$\text{PE} = \|\epsilon\| = \sqrt{\sum_{k=1}^4 [\Delta X S_{\text{in,def}}]_k^2}$$

as introduced in Equation 20 of (Ovelar et al., 2012).

2. Modulation–Demodulation Framework and the Response Matrix

A polarimeter’s signal chain is expressed in terms of a modulation matrix $O$ (dimension $m \times 4$ for $S_{\text{in}}^*$0 modulation states) and a corresponding demodulation matrix $S_{\text{in}}^*$1 ($S_{\text{in}}^*$2). Their product yields the system’s response matrix:

$S_{\text{in}}^*$3

Ideally, $S_{\text{in}}^*$4, but in the presence of non-idealities, perturbations are captured in $S_{\text{in}}^*$5:

$S_{\text{in}}^*$6

The measured Stokes vector after demodulation becomes $S_{\text{in}}^*$7. Perturbations impacting the response, including all optical-element-induced first-order variations and their propagation through the analysis chain, are integrated within $S_{\text{in}}^*$8.

3. Error Propagation: From Physical Parameter Errors to PE

Each optical element $S_{\text{in}}^*$9 is characterized by physical parameters $$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$0 with associated uncertainties $$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$1. The Mueller matrix for an element is decomposed as

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$2

Assuming small, independent errors, the total instrument Mueller matrix is approximated by expanding to first order in each $$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$3. The modulation matrix itself is likewise decomposed, and the demodulation matrix is computed as the Moore–Penrose pseudo-inverse of $$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$4 (Eq. 11):

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$5

Each perturbation term propagates to the response matrix via

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$6

The overall error in the output Stokes vector is the sum of errors from individual physical parameter perturbations acting on the design input Stokes vector:

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$7

Assuming mutual independence of the $$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$8, the total RMS error is

$$S_{\text{in}}^* = (I_4 + \Delta X) S_{\text{in,def}}$$9

For single-parameter contributions,

$\Delta X$0

and the total combined error is

$\Delta X$1

4. Underlying Assumptions and Approximations

The formalism underlying PE calculation is predicated on several explicit assumptions (as detailed in (Ovelar et al., 2012)):

• All physical-parameter errors $\Delta X$2 are sufficiently small for a first-order Taylor expansion to be accurate.
• The parameter uncertainties are statistically independent, justifying root-sum-square (RSS) combination.
• The demodulation step uses only the unperturbed modulation matrix $\Delta X$3 in its pseudo-inverse; inclusion of first-order terms in the inversion would yield non-linear $\Delta X$4 dependence and is not treated.
• The linearized Mueller calculus is assumed adequate for the full error budget.

This framework thus delivers a tractable, analytically transparent mapping from low-level physical tolerances to the high-level system error metric.

5. Workflow Implementation in Polarimetric Systems

The M&m's code operationalizes this formalism in the following stepwise workflow:

1. The optical train is specified as an ordered list of $\Delta X$5 elements, with associated physical parameters and uncertainties.
2. Element-wise, $\Delta X$6 and the set of first-order weight matrices $\Delta X$7 are provided.
3. The total system's “static” Mueller matrix is built via ordered multiplication of $\Delta X$8 for each modulation state.
4. In parallel, each $\Delta X$9 term is propagated as a first-order deviation, accounting for its correct position in the optical train.
5. First rows of each $4 \times 4$0 yield $4 \times 4$1 and the corresponding error-propagation matrices $4 \times 4$2.
6. Pseudo-inverse demodulation matrix $4 \times 4$3 is computed.
7. Each error contribution $4 \times 4$4 is formed, and all $4 \times 4$5 are summed to yield $4 \times 4$6.
8. For the design input Stokes vector, the error vector $4 \times 4$7 and PE are computed as

$4 \times 4$8

This modular architecture enables both single-parameter sensitivity studies and holistic error budgeting for complex systems.

6. Role and Interpretation in System Performance Analysis

PE serves as a scalar measure of polarimetric accuracy for a given input Stokes state. It can be directly juxtaposed with mission or science-driven requirements (e.g., stipulations such as "we require $4 \times 4$9 of full-scale") (Ovelar et al., 2012). Its utility extends to trade studies aimed at tolerance allocation: tighter requirements on $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$0 are imposed where their associated $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$1 contributions are dominant. Alternatively, analysis can be conducted over a suite of input states to determine the worst-case $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$2 or by comparison of $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$3 to a requirements matrix $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$4.

This approach formalizes traceability from physical parameter tolerances through to system-level polarimetric fidelity, supporting both design iteration and requirements verification.

7. Comparison with Alternative Metrics and Broader Implications

PE offers a single-number summary of system error, suitable for both specification compliance and optimization. While equally valid to directly analyze the full $$\epsilon = S_{\text{in}}^* - S_{\text{in,def}} = \Delta X S_{\text{in,def}}$$5 matrix or to explore state-dependent error structure, practice as documented in (Ovelar et al., 2012) deems PE the preferred figure of merit for most trade studies. Its analytical structure—clear partitioning of sources, explicit error propagation, and the ability to decompose by element or mechanism—provides actionable insight into instrument design. A plausible implication is that this metric is well-suited for both initial design phases and ongoing assessment as tolerances or requirements evolve.