Interferometry is the industry standard tool for measuring optics at the nanometer scale. It works because we are able to measure the phase of light using the equation for two-wave interference.

In this equation, a detector measures the irradiance $\text{(}I\text{)}$ on the left side of the equation. The right side of the equation describes two laser beams with irradiance $I_1$ and $I_2$. The phase of the first beam $\text{(}{\phi }_1\text{)}$ is a reference phase controlled by the instrument. Phase of the second beam $\text{(}{\phi }_2\text{)}$ is set by reflection from the optical surface being measured.

The goal of phase shifting is to estimate surface phase $\text{(}{\phi }_2\text{)}$ using detected irradiance $\text{(}I\text{)}$.

Algorithms

In practice, phase-shifting interferometers measure interference images at 3 or more discrete reference phase values. Three is the minimum number of measurements to eliminate the two laser irradiance values and still solve for the unknown surface phase.

The simulator below computes errors in surface phase $\text{(}{\phi }_2\text{)}$ created by instrument parameters: random irradiance noise, reference phase step amount, as well as random error in reference phase steps. Interference images assume red 633nm laser light and a surface with tilt, defocus, and spherical Zernike aberration.

Please note: this simulation incudes random values which change each time the page is loaded. To obtain a new set of values, please reload the page in your web browser (typical keyboard shortcut is Ctrl-R).

Although this simulator is limited to a small fraction of known phase shifting algorithms[1], they illustrate key features of phase shifting. Additional details are below.

Three Measurements

Irradiance measurements at 3 different reference phase values $\text{(}{\phi }_1\text{)}$ give 3 interference images with the same 3 unknowns $\text{(}{I}_1\text{, }{I}_2\text{, }{\phi }_2\text{)}$. Although 120° phase steps are sometimes used, 90° phase steps are conceptually easier to understand. A cosine wave phase shifted 90° becomes a sine wave.

The simplest mathematical solution uses phase shifts of [45°, 135°, 225°] shown above. Because overall phase is related to overall distance, adding 45° to every phase simplifies the calculation without changing any hardware. In other words, phase shifts of [45°, 135°, 225°] are experimentally the same as phase shifts of [0°, 90°, 180°].

Pulling everything together gives the following solution using 3 irradiance measurements $\text{(}{I}_a\text{, }{I}_b\text{, }{I}_c\text{)}$ from the 3 phase shifts. This equation can be applied pixel-by-pixel.

Four Measurements

Phase shifts can be created several ways. Traditionally, a piezo transducer was attached to a mirror in the reference beam path. Apply a voltage and the path length changes, thereby changing the phase $\text{(}{\phi }_1\text{)}$ of the reference beam. However, modern systems favor instantaneous phase shifting which uses polarization to measure multiple phases at the same time.

Consider two polarized light beams: one has right-hand circular polarization, and the other left-hand circular polarization. If both beams have equal irradiance, the sum of the two beams will be linearly polarized with orientation equal to twice the phase difference between the beams: $2({\phi }_2-{\phi }_1)$. Measuring irradiance values $\left(I_a\text{, }{I}_b\text{, }{I}_c\text{, }{I}_d\right)$ for linear polarization orientations [0°, 45°, 90°, 135°] is therefore equivalent to measuring phase shifts [0°, 90°, 180°, 270°]. The solution for these 4 phase measurements is given below.

The concept of polarization interferometry dates back to the early days of phase shifting interferometry[2,3], but products such as the Sony Polarsens camera[4] make this measurement trivial to implement. Using a grid of polarization filters over the sensor decreases spatial resolution, but the instantaneous measurement of all four phase shifts significantly reduces air turbulence and other time-varying phenomena.

Carré Equation (four measurements)

The earliest phase shifting algorithm[5] was published by P. Carré in 1966. Aside from its historical significance, this algorithm is also not limited to phase shifts which are multiples of 90°. Instead, it uses the extra 4^th measurement to eliminate the phase shift amount from the equations. You can see this in the simulation if you set the random phase and random irradiance to zero then change the constant phase step. Of the algorithms provided, only Carré algorithm gives zero error when the phase step changes.

Given four irradiance images $\left(I_a\text{, }{I}_b\text{, }{I}_c\text{, }{I}_d\right)$, the surface phase is computed as follows.

In practice, the Carré algorithm is not used much. Unless the phase steps are very close to perfect and noise is low, the advantages of this algorithm are minimal. Accurate cancellation requires all four phase shifts to be precise multiples of the same phase step.

Schwider-Hariharan (five measurements)

When phase shifting by a piezo, calibration and uncertainty of the phase shift amount is a significant source of error. If 1 volt is applied to a piezo, what exactly is the phase shift amount? Papers by Schwider et al.[6] and later by Harihan et al. [7] address this with clever averaging.

The concept is to extend the measurement by one step. If you have 4 phase steps [0°, 90°, 180°, 270°], then instead you should collect 5 irradiance values $\left(I_a\text{, }{I}_b\text{, }{I}_c\text{, }{I}_d\text{, }{I}_f\right)$ corresponding to phases [0°, 90°, 180°, 270°, 360°]. In essence, the first 4 measurements give you one phase estimate while the last 4 measurements give you a second phase estimate. This averaging helps reduce phase shift errors and random noise, while the processing still uses a relatively simple equation.

Because of its error suppression and simple calculation, the Schwider-Hariharan algorithm is good for temporal phase shifting interferometry.

Other Algorithms

This page only scratches the surface of phase shifting algorithms. Families of solutions are now known using algorithms with more than 100 phase steps[8], and all have their strength and weaknesses. Depending on the type of phase shifting used, and errors of greatest concern, there is likely an algorithm tailored for the need.

Final Comments

You may have noticed all the algorithms on this page involve the arctangent function. This resolves sign ambiguity from the cosine function in the equation for two-wave interference. However the output is still restricted in the range [−π, +π] radians. For phase values beyond this range, additional phase unwrapping is required. Those algorithms are beyond the scope of this page.

References

1. see chapter 14 of "Optical Shop Testing" 3rd ed., Malacara, John Wiley & Sons, 2007. https://doi.org/10.1002/9780470135976
2. "New Developments in Interferometry", part V "Interference Phase Measurement", Crane, Applied Optics, vol.8, 1969, pp.538-542. https://doi.org/10.1364/AO.8.000521
3. "Instantaneous phase measuring interferometry", Smythe and Moore, Optical Engineering, vol.23, 1984, pp.361-364. https://doi.org/10.1117/12.7973301
4. "Four-directional pixel-wise polarization CMOS image sensor using air-gap wire grid on 2.5-μm back-illuminated pixels", T. Yamazaki et al., IEEE International Electron Devices Meeting, 2016, pp. 8.7.1-8.7.4. https://doi.org/10.1109/IEDM.2016.7838378
5. "Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures", Carré, Metrologia, vol.2, 1966, pp. 13-23. https://doi.org/10.1088/0026-1394/2/1/005
6. "Digital wave-front measuring interferometry: some systematic error sources", Schwider et al., Applied Optics, vol.22, 1983, pp.3421-3432. https://doi.org/10.1364/AO.22.003421
7. "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm", Harihan, Oreb, and Eiju, Applied Optics, vol.26, 1987, pp.2504-2506. https://doi.org/10.1364/AO.26.002504
8. "101-frame algorithm for phase-shifting interferometry", de Groot, Proceedings SPIE, vol.3098, 1997, pp.283-292. https://doi.org/10.1117/12.281171