Interferometry Metrology Module

Surface flatness analysis for optical glass and precision flats measured on phase-shifting interferometers.

The module reads raw surface data straight out of the instrument (Zygo .datx and .dat, Apre and Zygo .xyz point exports), converts each into a common phase map in nanometres, removes the alignment terms, decomposes the residual onto a low-order Zernike basis, and reports the standard flatness parameters (PV, RMS, Power, Irregularity) in both nanometres and fringes.

A single class, VxManager, drives the whole pipeline so that every dataset and every instrument is processed identically.

Contents

• Installation
• Quick start
• Package layout
• Methodology
• Configuration reference
• Instrument formats
• Dummy
• Tests
• References

Installation

Python 3.12 or newer.

python -m venv .venv
.venv/Scripts/python.exe -m pip install numpy scipy h5py pandas matplotlib openpyxl

Package	Used for
numpy	array maths, least-squares solving
scipy	binary morphology for aperture edge trimming
h5py	reading the HDF5 container inside .datx
pandas, openpyxl	summary sheet export (.xlsx / .csv)
matplotlib	surface reports
pytest	optional, to run the test suite

Every instrument format is decoded in this module, so there is no third-party optics dependency.

Quick start

from Manager import VxManager

# .datx carries its own calibration
mgr = VxManager(trim=13)
results = mgr.Analyse("dataset.datx", report_dir="report")

for result in results:
    print(result.summary())

mgr.ExportSummary(results, "summary.xlsx")

# .xyz carries no calibration, so it is supplied through the config
mgr = VxManager(
    pixel_size_mm=0.07310224,
    wavelength_nm=633.0,
    waves_per_fringe=0.5,
    trim=13,
)
result = mgr.Analyse("2025-10-01_18-09-08_top.xyz")[0]
print(result.pv_fr, result.rms_fr, result.power_fr, result.irregularity_fr)

Batch a whole folder, rewriting the summary after every file so an interrupted run keeps its completed rows:

mgr.AnalyseFolder("datx", pattern="*.datx",
                  report_dir="report", summary_path="summary.xlsx")

Worked examples

The files in test/ are the worked example for each instrument as well as the test suite. Each one states that instrument's calibration as constants, runs the full pipeline on the matching dummy dataset, and can be run directly:

python test/test_zygo_datx.py         # writes reports and a summary sheet
python test/test_apre_xyz.py          # shows the surface report
python test/test_development_xyz.py   # top/bottom pair as surfaces 1 and 2

To analyse production data, point the *_DIR constant at the real folder or call VxManager directly as shown above.

Package layout

Manager/
  __init__.py        public API: VxManager, AnalysisConfig, Measurement, SurfaceResult
  manager.py         VxManager, the single orchestrator
  Models/            AnalysisConfig, Measurement, SurfaceResult
  Load/              extraction, one module per format
    datx.py          Zygo .datx (HDF5), ZygoExtract
    xyz.py           Apre / Zygo .xyz point exports, parsed in parallel
    dat.py           MetroPro binary .dat interferograms
    common.py        dataset naming, top/bottom pairing
  PreProcess/        coordinates, masking, edge trim, piston / tip-tilt / power removal
  Process/           PV, RMS, Power, Irregularity, modal fit
  Polynomials/       Zernike and Jacobi polynomials, least-squares mode fitting
  Plot/              surface report figure, summary sheet export

VxManager exposes the pipeline as five steps:

Method	Role
Load(path)	dispatches on file extension to _extractZygo, _extractApre or _extractXYZ; returns one Measurement per surface
Preprocess(m)	edge trim, then piston and tip/tilt removal, optional crop
Measure(m)	modal fit and the four flatness metrics
Analyse(path)	the three above, end to end, optionally writing a report
AnalyseFolder(dir)	Analyse over every matching file
Report, ExportSummary	figure and summary sheet output

Adding a format means adding one entry to VxManager._EXTRACTORS and one _extract* method.

Methodology

1. Extraction and unit calibration

Interferometers report optical path difference in fringes. A .datx file stores the scale factors alongside the data, and each surface of a stacked assembly is converted with its own calibration:

$$z_{\text{nm}} = N_{\text{fringes}} \cdot O \cdot S \cdot \lambda_{\text{nm}}$$

where $S$ is the interferometric scale factor (waves per fringe, normally 0.5 for a double-pass Fizeau cavity), $O$ the obliquity factor correcting for non-normal incidence, and $\lambda$ the measurement wavelength (632.8 nm for a HeNe source). The conversion follows the standard treatment of fringe order and optical path difference in interferogram analysis (Malacara, 2007; Wyant & Creath, 1992).

A .dat file is a big-endian MetroPro binary: a fixed header holding the camera regions and the calibration, then the intensity frames, then the phase map as 32-bit integer counts. Counts are decoded with the same relation, divided by the phase resolution the instrument digitised at,

$$z_{\text{nm}} = \frac{N_{\text{counts}} \cdot S \cdot O \cdot \lambda_{\text{nm}}}{R}, \qquad R \in {4096,; 32768,; 131072}$$

for 12, 15 and 17 bit resolution respectively, with counts at or above 2147483640 marking unmeasured pixels. Field offsets follow the MetroPro binary file format reference (Zygo Corporation, n.d.); the header also supplies the wavelength, the scale factor and the lateral resolution, so a .dat needs no configuration either.

.xyz exports carry no header calibration, so pixel size, wavelength and waves per fringe come from AnalysisConfig, and z_scale converts the file's own height unit into nanometres (Apre exports micrometres, so z_scale=1e3).

Point exports are parsed across worker processes and scattered onto a grid padded so that the measured aperture stays centred on the grid centre, which is where the Zernike pupil origin is placed in step 4.

2. Aperture conditioning

Invalid pixels ("No Data" dropouts, and everything outside the aperture) are held as NaN and excluded from every fit and statistic.

Edge pixels of a phase map are unreliable: the detector footprint straddles the physical edge of the part, so those pixels mix surface and background and bias PV strongly. The outer 13 pixel layers are therefore removed by iterated binary erosion,

$$M_{\text{trim}} = \left( \cdots \left( (M_{\text{filled}} \ominus B) \ominus B \right) \cdots \right) \cap M_{\text{valid}},$$

applied 13 times with the 4-connected structuring element $B$. Internal dropouts are filled before eroding, so only the outer boundary retreats and interior NaN islands do not grow. This is standard binary morphology (Serra, 1982; Haralick, Sternberg, & Zhuang, 1987).

3. Removing the alignment terms

Piston (a constant offset) and tip/tilt (a rigid rotation of the part in the cavity) are artefacts of alignment, not of the surface, and are removed before any metric is computed:

$$z \leftarrow z - \bar{z}, \qquad z \leftarrow z - (a x + b y),$$

with $a, b$ from the least-squares plane fit

$$\min_{a,b} \sum_{i \in \text{valid}} \left( a x_i + b y_i - z_i \right)^2 .$$

All fits in the module use the same routine, Polynomials.lstsq, which solves the normal equations through the singular value decomposition provided by numpy.linalg.lstsq, giving the minimum-norm solution even when the design matrix is rank deficient (Golub & Van Loan, 2013).

4. Modal decomposition on a Zernike basis

The conditioned surface is projected onto the first four Zernike polynomials in Fringe (University of Arizona) ordering, $Z_1$ to $Z_4$, that is piston, tip, tilt and defocus (Wyant & Creath, 1992). Zernike polynomials are the natural basis here because they are orthogonal on the unit disc, which matches a circular optical aperture, and because their low-order terms correspond directly to the recognised aberrations (Zernike, 1934; Born & Wolf, 1999; Noll, 1976).

In polar coordinates $(\rho, \theta)$ on a pupil normalised so that $\rho = 1$ at the outermost valid pixel,

$$Z_n^m(\rho, \theta) = R_n^{|m|}(\rho) \cdot \begin{cases} \cos(m\theta), & m \ge 0 \\ \sin(|m|\theta), & m < 0 \end{cases}$$

$$R_n^{|m|}(\rho) = \sum_{k=0}^{(n-|m|)/2} \frac{(-1)^k , (n-k)!}{k! \left( \frac{n+|m|}{2} - k \right)! \left( \frac{n-|m|}{2} - k \right)!} \rho^{,n-2k}$$

The factorial form above is implemented directly in Polynomials.zernike_radial, but it is numerically poor at high order and recomputes shared work for every mode. The sequence generator used by the pipeline instead exploits the identity between the Zernike radial polynomials and the Jacobi polynomials $P_k^{(\alpha, \beta)}$,

$$R_n^{|m|}(\rho) = \rho^{|m|} , P_{(n-|m|)/2}^{(0,, |m|)}!\left( 2\rho^2 - 1 \right),$$

and evaluates the Jacobi polynomials by their three-term recurrence

$$P_{k+1} = (A_k x + B_k) P_k - C_k P_{k-1}$$

with the coefficients of Abramowitz and Stegun (1964, ch. 22) and the NIST Digital Library of Mathematical Functions (Olver et al., n.d., §18.9). For a sequence of modes this evaluates one Jacobi recurrence per $|m|$ and reuses $\rho^{|m|}$, $\sin(|m|\theta)$ and $\cos(|m|\theta)$ across every radial order, rather than recomputing a factorial sum per mode. Recurrence-based evaluation of Zernike terms is the established approach for this reason (Prata & Rusch, 1989; Shakibaei & Paramesran, 2013).

Modes are generated unnormalised for the metric basis, so each has zero-to-peak amplitude 1 and the fitted coefficients read directly in nanometres of surface. The orthonormal scaling

$$N_n^m = \sqrt{\frac{2(n+1)}{1 + \delta_{m0}}}$$

is available through norm=True when unit-RMS modes are wanted instead (Noll, 1976).

The coefficient vector is obtained by least squares over the valid pixels only:

$$\mathbf{c} = \arg\min_{\mathbf{c}} \left| \mathbf{A}\mathbf{c} - \mathbf{z} \right|_2^2, \qquad \mathbf{A}_{ij} = Z_j(\rho_i, \theta_i).$$

5. Flatness metrics

Metric	Definition	Basis
PV	$\max z - \min z$	peak-to-valley of the surface
RMS	$\sqrt{\frac{1}{N} \sum_i z_i^2}$	areal $S_q$ (ISO 25178-2:2012)
Power	$2 c_4$	peak-to-valley of the best-fit sphere
Irregularity	$\max r - \min r$, $; r = z - \mathbf{A}\mathbf{c}$	PV of the residual after $Z_1$ to $Z_4$ (ISO 10110-14)

Power deserves a note. With unnormalised modes the defocus term is $Z_4 = 2\rho^2 - 1$, which spans $[-1, +1]$ across the pupil, so a fitted coefficient $c_4$ contributes a peak-to-valley of exactly $2 c_4$. Reporting $2 c_4$ therefore states the power as the peak-to-valley sag of the best-fit sphere, which is the convention used by commercial interferometer software and by the surface form tolerances of ISO 10110-5:2015.

Irregularity separates the two failure modes that PV alone conflates. A part may be out of tolerance because it is spherically bowed (power, often a polishing or mounting effect and sometimes correctable) or because it is locally rough and astigmatic (irregularity, which is not). Subtracting the fitted piston, tilt and defocus leaves exactly the second component, following the wavefront deformation tolerances of ISO 10110-14 (Malacara, 2007).

6. Fringe normalisation

Every metric is also reported in fringes, so that results are comparable across instruments and wavelengths:

$$v_{\text{fr}} = \frac{v_{\text{nm}}}{\lambda_{\text{nm}} \cdot S}$$

With $S = 0.5$ the denominator is $\lambda/2$, the surface height change that advances a double-pass interferogram by one fringe.

Configuration reference

AnalysisConfig, also settable as keyword arguments to VxManager:

Field	Default	Meaning
pixel_size_mm	None	lateral sampling; required for .xyz, read from file for .datx
wavelength_nm	633.0	measurement wavelength; read from file for .datx
waves_per_fringe	0.5	interferometric scale factor $S$
z_scale	1.0	factor converting the file's height unit to nm (1e3 for micrometres)
swap_xy	False	set when an .xyz export lists row before column
trim	13	aperture edge layers removed; 0 disables
crop	False	crop to the valid bounding box after form removal
n_workers	8	processes used to parse .xyz files

Instrument formats

Extension	Extractor	Container	Calibration	Surfaces per file
.datx	_extractZygo	HDF5	from the file	one per stacked assembly entry
.dat	_extractApre	MetroPro binary	from the file	one
.xyz	_extractXYZ	text point list	from the config	one

Dummy

dummy dataset/ holds three small synthetic datasets per instrument format, so the pipeline can be run without production data.

Folder	Datasets	Files	Type
apre/	3	6	S1 to S3, each as a dense .xyz in micrometres and as a MetroPro .dat
zygo/	3	3	Zygo .datx, a stacked assembly of 2 surfaces each, so 6 surfaces in total
development xyz/	3	6	development-set .xyz, one top and one bottom export per dataset

Every surface is a circular aperture on a 96 x 80 grid. The development-set exports are partial and start at index 6, so they pad to 108 x 92 when loaded, which is what exercises the centred-padding path. Regenerate with:

python "dummy dataset/generate.py"

The surfaces are analytic (tilt, defocus, astigmatism, ripple and noise), so the metrics are repeatable but not physically meaningful: they exist to exercise the readers and the pipeline, not to validate against instrument results.

Tests

.venv/Scripts/python.exe -m pytest test -q

One module per source format, each trimming the same 13 pixel layers, and each also runnable on its own as the worked example for that instrument:

File	Covers
test/test_zygo_datx.py	calibration read from file, stacked surfaces, trim depth, piston and tilt removal, report and summary output, folder batching
test/test_apre_xyz.py	config-supplied calibration, pixel spacing, micrometre to nanometre scaling, dense grid with No Data, crop path
test/test_development_xyz.py	partial export padded to keep the aperture centred, top/bottom pairing, summary column layout, .xlsx and .csv agreement

Each file declares its instrument's wavelength, waves per fringe and pixel size as constants, so the calibration used for a given source is stated in one place and asserted on every run.

References

Methodology

Abramowitz, M., & Stegun, I. A. (Eds.). (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables (Applied Mathematics Series 55). National Bureau of Standards.

Born, M., & Wolf, E. (1999). Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (7th ed.). Cambridge University Press.

Creath, K. (1988). Phase-measurement interferometry techniques. In E. Wolf (Ed.), Progress in optics (Vol. 26, pp. 349-393). Elsevier.

Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed.). Johns Hopkins University Press.

Haralick, R. M., Sternberg, S. R., & Zhuang, X. (1987). Image analysis using mathematical morphology. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-9(4), 532-550.

International Organization for Standardization. (2003). Optics and photonics: Preparation of drawings for optical elements and systems - Part 14: Wavefront deformation tolerance (ISO 10110-14:2003).

International Organization for Standardization. (2012). Geometrical product specifications (GPS): Surface texture: Areal - Part 2: Terms, definitions and surface texture parameters (ISO 25178-2:2012).

International Organization for Standardization. (2015). Optics and photonics: Preparation of drawings for optical elements and systems - Part 5: Surface form tolerances (ISO 10110-5:2015).

Malacara, D. (Ed.). (2007). Optical shop testing (3rd ed.). John Wiley & Sons.

Noll, R. J. (1976). Zernike polynomials and atmospheric turbulence. Journal of the Optical Society of America, 66(3), 207-211.

Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S., & McClain, M. A. (Eds.). (n.d.). NIST digital library of mathematical functions. Retrieved from https://dlmf.nist.gov/18.9

Prata, A., & Rusch, W. V. T. (1989). Algorithm for computation of Zernike polynomials expansion coefficients. Applied Optics, 28(4), 749-754.

Serra, J. (1982). Image analysis and mathematical morphology. Academic Press.

Shakibaei, B. H., & Paramesran, R. (2013). Recursive formula to compute Zernike radial polynomials. Optics Letters, 38(14), 2487-2489.

Wyant, J. C., & Creath, K. (1992). Basic wavefront aberration theory for optical metrology. In R. R. Shannon & J. C. Wyant (Eds.), Applied optics and optical engineering (Vol. 11, pp. 1-53). Academic Press.

Zernike, F. (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica, 1(7-12), 689-704.

Zygo Corporation. (n.d.). MetroPro reference guide: Binary data file formats [Software documentation].

Software

Harris, C. R., Millman, K. J., van der Walt, S. J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N. J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M. H., Brett, M., Haldane, A., del Rio, J. F., Wiebe, M., Peterson, P., ... Oliphant, T. E. (2020). Array programming with NumPy. Nature, 585(7825), 357-362. https://doi.org/10.1038/s41586-020-2649-2

Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in Science & Engineering, 9(3), 90-95. https://doi.org/10.1109/MCSE.2007.55

Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., ... SciPy 1.0 Contributors. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 17(3), 261-272. https://doi.org/10.1038/s41592-019-0686-2

A note on standard editions

The ISO editions cited are those referenced during development. Confirm the currently published edition before quoting a tolerance in a drawing or an inspection report.