Objective  Traditional single-wavelength interferometry is not suitable to unwrap the correct phase for measuring surface with step or groove, whose depth is larger than half wavelength. Dual-wavelength interferometry (DWI) technique employs an extra wavelength to generate a longer beat-frequency synthetic wavelength (\lambda _\rms). For synthetic wavelength is much longer than the optical working wavelength, DWI extends the measuring discontinuity limit of interferometry greatly. And DWI could achieve the simultaneous accurate measurements with large dynamic range for the multi-scale morphology characteristics parameters such as the macro profile and local morphology with step. Meanwhile, in the simultaneous dual-wavelength interferometry (SDWI), the two single-wavelength interferogram is captured simultaneously to accelerate the data collection, which is immune to vibration with the advantages of the time saving and higher efficiency. In practice, the dual-wavelength interferogram is usually captured by the monochrome sensor, which is more convenient and economical. And a generated dual-wavelength Moiré fringe pattern appears as the simple incoherent additive superposition of the two corresponding single-wavelength interferogram. The low beat-frequency envelope of the generated fringe pattern indirectly represents the needed synthetic-wavelength information, whose direct extraction is rather arduous. For this purpose, we present a dual-wavelength interferometric algorithm combining with spatial-temporal conjugate complex functions coupling and double phase shift strategy.

  Methods  The method needs to acquire multiple groups of phase-shift dual-wavelength interferogram, and each group consists of four continuous interferogram. The phase shift step among the four frames in each group is required as π/2 at one single wavelength, while π/2 at synthetic wavelength between the adjacent groups by the designed double phase shift strategy (Fig.2). And in dual-wavelength squeezing interferometry for each group, the temporal phase shift in each group is converted into spatial carrier in the generated dual-wavelength STF. Therefore, the +1-order spectral lobes for the two wavelengths could be easily separated from others and filtered in the Fourier spectrum of the generated dual-wavelength STF without extra spatial carrier and elimination of background. After the appropriate band-pass filter and inverse Fourier transform, the single-wavelength interferometric complex functions are obtained. Subsequently, when the conjugate single-wavelength interferometric complex functions are multiplied, the synthetic-wavelength interferometric fringe pattern could be extracted directly (Fig.1). The obtained synthetic-wavelength interferogram from each group with π/2 phase-shift step at \lambda _\rms could be demodulated to retrieve the final synthetic-wavelength phase by the conventional phase-shift algorithm.

  Results and Discussions  Simulations verify that the proposed method has a lower linear carrier requisition than the spatial-domain Fourier transform demodulation theory, which is merely about 0.077 times of latter numerically, even the phase-shift deviation at different wavelength exists (Fig.4). Besides, the feasibility and applicability of the proposed method are verified using simulation and experimental results. Numerical simulation indicates that the demodulated error is better than PV of 0.556 9 nm and RMS of 0.089 7 nm even when the fringe number is 1 at \lambda _\rms (Fig.3). In addition, for the test step in the experiment, the results have validated the effectiveness of the proposed method for the interferogram with lower linear carrier. And the step height deviations for the proposed method are better than 0.94% for the step with the height of 7.8 μm even for one fringe at \lambda _\rms (Fig.9).

  Conclusions  To extract and retrieve the lower frequency synthetic-wavelength interferometric fringe form dual-wavelength Moiré fringe, we present a dual-wavelength interferometric algorithm combining with spatial-temporal conjugate complex functions coupling and double phase shift strategy. Several groups of phase-shift dual-wavelength interferogram are acquired with every contiguous four frames in each group. The temporal phase shift among each group is converted into spatial carrier for the spectral separation with lower spatial carrier. When the obtained spatial-temporal conjugate complex functions are coupling by multiplication, one synthetic-wavelength interferogram could be extracted for each group directly without the introduction of other wavelength interferometric information. For the designed π/2 phase shift at synthetic wavelength between the adjacent groups, the extracted synthetic-wavelength interferogram from every group is demodulated by conventional phase-shift algorithm directly. The proposed method has a lower carrier requisition than the traditional spatial-domain Fourier demodulation theory, which is merely about 0.077 times of the former numerically, even when the phase-shift deviation for different wavelengths exists.