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Phase-shifting interferometry

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Classic interference fringes on a flat surface under monochromatic light.[1]

Phase-shifting interferometry (PSI)[2][3] is an optical metrology technique for measuring surface topography and wavefront shape to sub-nanometre precision.[4] It works by recording a short sequence of interferograms, each captured with a known phase offset between the reference and object beams, then solving for the surface height at every pixel simultaneously.[5][6] Height measurement repeatability is typically below 1 nm, independent of field size.[1] Before PSI, the standard approach was to trace fringe centres in a static interferogram by eye or with image-processing software, which was a labour-intensive, sparse, and unable to resolve the sign of surface deviations from a single image.[7]

Principle

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A Twyman–Green interferometer configured as a white-light scanner. In a PSI measurement the reference mirror is stepped by a piezoelectric actuator between frames rather than scanned continuously.[1]

The phase shift is most often introduced by a piezoelectric transducer (PZT) moving the reference mirror in steps of roughly a quarter wavelength of optical path between frames.[8][9] With at least three frames, the three unknowns at each pixel namely, background intensity, fringe contrast, and surface phase, are over-determined and can be solved with an arctangent formula.[10][11] The raw output is a wrapped phase map; a phase-unwrapping[12] step removes the 2π jumps to give the final height map.[13][14] PSI can be implemented in Twyman–Green, Fizeau, Mach–Zehnder, and common-path configurations.[8]

History

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P. Carré described a four-frame algorithm tolerant of unknown step sizes as early as 1966.[15][16] The pivotal step toward practical use came in 1974, when Bruning and colleagues at Bell Laboratories built a fully digital, computer-controlled system for testing optical surfaces and lenses.[17] This is often cited as the birth of modern PSI.[6] Widespread industrial adoption came in the 1980s as affordable CCD arrays and desktop computers made real-time phase calculation possible.[10] Comprehensive reviews by Creath (1988) and Greivenkamp and Bruning (1992) cemented PSI as the standard framework for precision interferometric metrology.[18][19]

The four-step algorithm with π/2 steps is the most common basic form.[10] Schwider and colleagues (1983) showed that a five-step variant largely cancels errors from PZT miscalibration.[20] Hariharan, Oreb, and Eiju (1987) independently derived the same formula, now known as the Schwider–Hariharan algorithm.[21] The main remaining error sources are mechanical vibration during acquisition, detector nonlinearity, and stray reflections; higher-order algorithms and simultaneous multi-channel designs have been developed to address each of these.[22]

Applications

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Common uses include optical component testing, semiconductor wafer and MEMS characterisation, and precision-engineering quality control.[1][23]

References

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  1. 1 2 3 4 de Groot, P. (2011). "Phase Shifting Interferometry". In Leach, R. (ed.). Optical Measurement of Surface Topography. Berlin: Springer. pp. 167–186. doi:10.1007/978-3-642-12012-1_8. ISBN 978-3-642-12011-4.
  2. ↑ Schreiber, Horst; Bruning, John H. (2007-06-08), "Phase Shifting Interferometry", in Malacara, Daniel (ed.), Optical Shop Testing (1 ed.), Wiley, pp. 547–666, doi:10.1002/9780470135976.ch14, ISBN 978-0-471-48404-2, retrieved 2026-05-12
  3. ↑ "Phase-Shifting Interferometry". James C. Wyant. Retrieved 2026-05-15.
  4. ↑ Saltik, Alperen; Saylan, Sueda; Tokel, Onur (2024). "Fourier-transform-only method for random phase shifting interferometry". Journal of Optics. 26 (3): 035604. Bibcode:2024JOpt...26c5604S. doi:10.1088/2040-8986/ad237c. ISSN 0150-536X.
  5. ↑ Song, Shijun; Liu, Xinyue; Chen, Tao; Liu, Changhua; An, Qichang (2025-11-29). "Principles and Applications of Interferometry in Highly Segmented Mirrors Co-Phasing". Photonics. 12 (12): 1181. Bibcode:2025Photo..12.1181S. doi:10.3390/photonics12121181. ISSN 2304-6732.
  6. 1 2 Bruning, J. H.; Herriott, D. R.; Gallagher, J. E.; et al. (1974). "Digital wavefront measuring interferometer for testing optical surfaces and lenses". Applied Optics. 13 (11): 2693–2703. Bibcode:1974ApOpt..13.2693B. doi:10.1364/AO.13.002693. PMID 20134757.
  7. ↑ "Phase-Shifting Interferometry for Determining Optical Surface Quality". Newport Corporation. Retrieved 2026-05-01.
  8. 1 2 Servin, M.; Estrada, J. C.; Quiroga, J. A. (2020). "Phase-shift interferometry". Journal of Optics. 22 (10): 103501. doi:10.1088/2040-8986/abb1d1.
  9. ↑ Machuca-Bautista, Yanely B.; Strojnik, Marija; Flores, Jorge L.; Serrano-García, David I.; García-Torales, Guillermo (2021-12-01). "Michelson interferometer for phase shifting interferometry with a liquid crystal retarder". Results in Optics. 5 100197. Bibcode:2021ResOp...500197M. doi:10.1016/j.rio.2021.100197. ISSN 2666-9501.
  10. 1 2 3 Wyant, J. C. "Phase Shifting Interferometry" (PDF). University of Arizona. Retrieved 2025-07-18.
  11. ↑ "Phase shifting techniques [Interference and Diffraction]". optique-ingenieur.org. Retrieved 2026-05-12.
  12. ↑ Robinson, Simon; Schödl, Horst; Trattnig, Siegfried (2014). "A method for unwrapping highly wrapped multi-echo phase images at very high field: UMPIRE". Magnetic Resonance in Medicine. 72 (1): 80–92. doi:10.1002/mrm.24897. ISSN 1522-2594. PMC 4062430. PMID 23901001.
  13. ↑ "Phase Wrapping - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2026-05-12.
  14. ↑ "OPG". opg.optica.org. Retrieved 2026-05-12.
  15. ↑ Carré, P. (1966). "Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures". Metrologia. 2 (1): 13–23. Bibcode:1966Metro...2...13C. doi:10.1088/0026-1394/2/1/005.
  16. ↑ Muhamedsalih, Hussam; Tang, Dawei; Kumar, Prashant; Jiang, Xiangqian (2022-02-05). "Carré Phase Shifting Algorithm for Wavelength Scanning Interferometry". Machines. 10 (2): 116. doi:10.3390/machines10020116. ISSN 2075-1702.
  17. ↑ "The Race to Electronic Photography". www.optica-opn.org. Retrieved 2026-05-12.
  18. ↑ Creath, K. (1988). "Phase-Measurement Interferometry Techniques". In Wolf, E. (ed.). Progress in Optics. Vol. 26. Amsterdam: Elsevier. pp. 349–393.
  19. ↑ Greivenkamp, J. E.; Bruning, J. H. (1992). "Phase shifting interferometry". In Malacara, D. (ed.). Optical Shop Testing (2nd ed.). New York: Wiley. pp. 501–598.
  20. ↑ Schwider, J.; Burow, R.; Elssner, K. E.; Grzanna, J.; Spolaczyk, R.; Merkel, K. (1983). "OPG". Applied Optics. 22 (21): 3421. doi:10.1364/AO.22.003421. PMID 18200214. Retrieved 2026-05-12.
  21. ↑ Hariharan, P.; Oreb, B. F.; Eiju, T. (1987). "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm". Applied Optics. 26 (13): 2504–2506. Bibcode:1987ApOpt..26.2504H. doi:10.1364/AO.26.002504. PMID 20489904.
  22. ↑ de Groot, P. (1995). "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window". Applied Optics. 34 (22): 4723–4730. Bibcode:1995ApOpt..34.4723D. doi:10.1364/AO.34.004723. PMID 21052308.
  23. ↑ Kujawinska, M.; Malinowski, M.; Malinowska, K. (2015). Phase Measurement Techniques. Bellingham: SPIE.
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