How to Set Up Pass/Fail Criteria for Cylinder Axis

The Physics of Tolerance and The “Sliding Scale”

Setting quality control limits for Sphere Power is easy: it is a linear scalar. If the spec is ±0.25D, the logic is binary. Setting limits for Cylinder Axis, however, is one of the most complex challenges in optical manufacturing.

• The Conflict: The tolerance is not static. It depends entirely on the magnitude of the Cylinder Power.
• The Physics: A 5° error on a low-cylinder lens is cosmetic. A 5° error on a high-cylinder lens is catastrophic.
• The Trap: Applying a “blanket tolerance” (e.g., ±5° for all SKUs) guarantees either massive yield loss (on low cyls) or massive customer complaints (on high cyls).

This section establishes the mathematical and physical foundation for a dynamic Pass/Fail logic based on Vector Addition and ISO 18369 protocols.

The “Resultant Cylinder” Concept

To understand why Axis criteria must be dynamic, we must look at the Induced Cylinder.

When a Toric lens is rotated off-axis by an angle, it does not just fail to correct the patient’s astigmatism; it introduces a new residual astigmatism vector.

The Vector Approximation: For a lens with Cylinder Power C and an axis misalignment δ (in radians), the magnitude of the unwanted residual cylinder (C_res) is:

C_res ≈ 2 · C · sin(δ)

The “Blur Cost” Analysis: Let’s define the “Blur Threshold” visible to a sensitive patient as 0.25D. We can calculate the maximum allowable axis error (δ_max) for different lens powers to stay under this threshold.

Low Cyl Lens (C = -0.75D): 0.25 = 2 · 0.75 · sin(δ) sin(δ) = 0.166 → δ ≈ 9.6° Result: We can be sloppy. A 9° error is acceptable.

High Cyl Lens (C = -2.75D): 0.25 = 2 · 2.75 · sin(δ) sin(δ) = 0.045 → δ ≈ 2.6° Result: We must be precise. A 3° error is a failure.

Conclusion: Pass/Fail criteria cannot be a single number. They must be a Function f(C).

The ISO 18369-2 Staircase

The International Standard Organization (ISO) codified this physics into a step-function tolerance. When setting up your MES (Manufacturing Execution System) or Metrology software, you cannot hard-code a single limit. You must program a Lookup Table.

The Standard Limits (ISO 18369-2:2017):

Cylinder Power Range (F’c)	Axis Tolerance (θ_tol)
0.00 to 0.50 D	± 10° (or undefined for <0.25D)
0.51 to 1.50 D	± 5°
1.51 to 2.50 D	± 4°
> 2.50 D	± 3°

The Process Engineering Challenge:

While ISO defines the market limit, the production limit must be tighter. This is due to Measurement Uncertainty.

If your Metrology Gauge R&R is ± 1.0°, and you set your Pass/Fail limit at exactly ± 3.0°, you will ship bad lenses (Type II Error) and scrap good lenses (Type I Error).

The Low-Cylinder Singularity

A common error in setting up Pass/Fail criteria is failing to handle the Near-Zero Cylinder condition.

• The Math: As Cylinder Power (C) approaches 0, the Axis (θ) becomes mathematically undefined.
• The Artifact: In a Moiré Deflectometer or Focimeter, a perfectly spherical lens (C=0) is never measured as 0.000. Noise will result in a reading of C=0.02D with a random Axis (e.g., 45°).
• The Logic: Your Pass/Fail logic must include a “Cyl Threshold Gate”.
  • IF Measured_Cyl < 0.25D THEN Axis_Result = “N/A” (Pass).
  • IF Measured_Cyl ≥ 0.25D THEN Evaluate Axis Tolerance.

Without this gate, your production yield for Spherical lenses (which often have a tiny residual cyl) will crash because they will randomly “fail” an axis check that shouldn’t exist.

The foundation of Axis criteria is the realization that Axis is a multiplier, not an additive. The error scales with the power. Therefore, the QA architecture must be built on a “Sliding Scale” logic. A system that flags a 4° error on a -0.75D lens as a “Fail” is poorly engineered. A system that passes a 4° error on a -3.00D lens is negligent.

In Part 2, we will explore the Statistical Implementation: How to calculate Cpk for circular data and how to handle the “Wrap-Around” problem of 0/180 degrees.

Statistics, Guard Banding, and Circular Logic

In Part 1, we defined the physical limits. In Part 2, we discuss the implementation.

Setting up a Pass/Fail criterion in an automated line involves statistical process control (SPC). However, standard statistical tools (Excel, Minitab) often fail with Axis data because degrees are Circular, not Linear.

Furthermore, we must define the Guard Band-the safety margin we subtract from the ISO limit to ensure compliance.

The “Wrap-Around” Problem (0° vs 180°)

The Cylinder Axis is periodic over 180°. 0° ≡ 180°.

The Data Trap: Imagine a batch of 5 lenses targeting 180°.

• Readings: 179°, 178°, 1°, 2°, 180°.
• Linear Average: (179 + 178 + 1 + 2 + 180) / 5 = 108°. (Massive Failure)
• True Average: 0° (or 180°). (Perfect Batch)

The Fix: Minimized Angular Difference Your Pass/Fail algorithm must never subtract raw angles. It must calculate the Shortest Arc.

Δθ = min(|θ_meas – θ_target|, 180 – |θ_meas – θ_target|)

QA Rule: Ensure your SPC software (or Excel macro) utilizes vector averaging or modulo arithmetic; otherwise, your Cpk calculations for “With-the-Rule” astigmatism (180°) will be mathematically garbage.

Establishing the “Guard Band” (TUR)

You cannot set the production limit equal to the ISO limit. You must account for Test Uncertainty Ratio (TUR).

Formula for Production Limit (L_prod):

L_prod = L_ISO – (U_metrology × k)

Where:

• L_ISO is the Standard Limit (e.g., 5°).
• U_metrology is the Expanded Uncertainty of your sensor (e.g., 0.8°).
• k is a risk factor (usually 1.0 to 1.6).

Example Scenario (Cyl = -1.25D):

• ISO Limit: ± 5.0°.
• Machine Repeatability (3σ): ± 1.0°.
• Fiducial Alignment Error: ± 0.5°.
• Total Uncertainty: √(1.0² + 0.5²) ≈ 1.1°.
• Production Limit: 5.0° – 1.1° = ± 3.9°.

Yield Consequence: By setting the limit to 3.9°, you are scrapping lenses between 3.9° and 5.0° that might be good. This is the cost of metrology uncertainty. To improve yield, you don’t widen the limit; you improve the metrology (better loading, better algorithms) to reduce U and reclaim the band.

The “Double Standard” for Axis

In high-volume manufacturing, we often split the criteria into Statistical Control Limits (for the machine) and Release Limits (for the product).

1. Process Control Limits (UCL/LCL):

• Used to monitor the Mold Alignment.
• Typical limit: ± 2.0°.
• Logic: If the average axis shifts by 2°, the mold keying is wearing out. Stop the line. Even if the lenses are passing ISO (± 5°), the process is drifting.

1. Product Release Limits:

• Used to accept/reject individual lenses.
• Typical limit: Guard-Banded ISO (e.g., ± 3.9°).

2.4 Handling “Fiducial Offset” Calibration

Every Toric mold has a systematic offset. The mechanical keying feature (datum) is never perfectly aligned with the optical toric curve (θ_opt).

The Offset Parameter:

Offset = θ_target – θ_measured_average

The Setup Procedure:

1. Run a sample batch (N=30) of a master Toric design.
2. Calculate the vector average of the axis error.
3. If the batch average is +2.5° (systematic), do not reject the batch.
4. Action: Input a -2.5° “Software Offset” into the metrology recipe.
5. Criteria: The Pass/Fail limit applies to the Random Variation (σ), not the systematic bias (which is calibratable).

Warning: Only apply offsets if the bias is stable (mold specific). If the bias fluctuates, it is a process failure.

Setting criteria requires a statistical architecture that respects the circular nature of angles and the uncertainty of the test method.

The “Guard Band” is the financial lever: widen it, and you risk recalls; narrow it, and you burn money.

In Part 3, we will look at Dynamic Retesting Strategies-how to handle the “Grey Zone” lenses that sit right on the edge of the Pass/Fail limit.

The “Grey Zone” Strategy and Dynamic Logic

In the final section, we move from theory to the factory floor logic. What happens when a lens measures exactly at the limit?

Because Axis measurement is highly sensitive to lens loading (tilt/rotation), a single “Fail” result is often a “False Fail.”

This section outlines a Dynamic Retest Logic (Conditional Logic) that maximizes yield without compromising quality.

The Three-Zone Logic

Instead of a binary Pass/Fail, modern metrology systems utilize three zones:

1. Green Zone (Pass): Result is within the “Safe Limit” (e.g., < ± 3.0°).
   • Action: Release.
2. Red Zone (Hard Fail): Result is gross failure (e.g., > ± 8.0°).
   • Action: Scrap immediately. (Likely a mold rotation or inverted lens).
3. Yellow Zone (Retest/Grey): Result is between Safe and ISO Limit (e.g., 3.0° to 5.0°).
   • Action: Trigger Retest Logic.

Designing the Retest Algorithm

Why retest? Because soft lenses are flexible. A “Yellow Zone” result is often due to a loading error (tweezer stress, bubble, or tilted load) rather than a manufacturing defect.

The “2 out of 3” Rule: If a lens falls in the Yellow Zone:

1. Unload the lens.
2. Reload (allow relaxation time).
3. Measure Again.
   • If Result #2 is Green → PASS. (Assume Result #1 was a loading error).
   • If Result #2 is Yellow/Red → FAIL.

The “Vector Averaging” Rule (Advanced): For high-value lenses, take 3 measurements.

• Do not average the scalars.
• Convert (Cyl_1, Axis_1), (Cyl_2, Axis_2), (Cyl_3, Axis_3) to J0/J45 vectors.
• Calculate the Mean Vector.
• Check if the Mean Vector falls within the ellipse of tolerance.
• Benefit: This filters out random noise (loading jitter) and reveals the true geometric axis of the lens.

Dynamic Tolerance based on SNR (Signal-to-Noise)

Not all measurements are equal. A Deflectometer can calculate a “Confidence Score” or SNR based on the contrast of the Moiré fringes.

Logic:

• If SNR > 90% (Clean Signal) → Use Tight Limit (± 4.0°).
• If SNR < 60% (Noisy Signal/Bubble) → The Axis reading is unreliable.
• Action: Do not fail the lens on Axis. Flag “Measurement Invalid” and force a reload.
• Benefit: Prevents scrapping good lenses due to dirty saline or bubbles.

Handling Multi-Cavity Molds

A critical failure mode in Toric manufacturing is when the Toric feature “washes out” (e.g., incomplete mold fill).

• The Symptom: Cylinder Power drops from -1.25D to -0.50D.
• The Axis: The Axis might still be perfect (0° error).
• The Pass/Fail Trap: A simple system checks Axis (0° = Pass) and checks Cyl (-0.50D = Fail?).
• The Criteria Link: You must link the criteria.
  • Rule: “Axis Result is only Valid if Cylinder Power is within ± 0.25D of Target.”

If Cylinder Power is low, the Axis reading is physically meaningless. Fail the lens on “Power,” do not confuse the data with “Axis” statistics.

Summary Table: Setting the Logic

Parameter	Criteria Setting	Rationale
Logic Type	Sliding Scale (by Power)	Follows ISO 18369 physics.
Zero-Handling	Modulo 180°	Handles 179°/1° averaging.
Guard Band	ISO Limit – 1.5 × U_gauge	Accounts for metrology error.
Low Cyl Gate	If Cyl < 0.25D, Axis = N/A	Prevents noise-based failures.
Retest	Yellow Zone Logic	Filters loading errors.
Bias	Cavity-Specific Offsets	Calibrates systematic tool errors.

Conclusion

Setting Pass/Fail criteria for Cylinder Axis is not a data-entry task; it is a systems engineering task. It requires understanding that the Axis is Dynamic (depends on Power), Circular (wraps at 180), and Sensitive (prone to loading noise).

By moving away from static “± 5 degrees” blankets and adopting Power-Dependent Guard Banding and Vector-Based Logic, manufacturers can stop fighting their metrology equipment and start controlling their true optical quality. The goal is not just to pass ISO; it is to ensure that the patient, whose vision depends on that vector alignment, sees the world clearly.

Disclaimer: 

This document is intended for educational use only. It does not represent legal, regulatory, or certification advice, and should not be interpreted as a declaration of compliance or approval by Rotlex or any regulatory authority.