Evaluation of spectral modulated interferograms using a Fourier transform and the iterative phase-locked loop method

Abstract

Spectral modulated interference fringes are observed in the form of the periodical modulation of a broadband spectrum at the output of an interferometer provided with a subsequent spectrometer. The group optical path difference of interfering light waves corresponding to the distance from the surface to be measured is characterized by the spectral fringe phase function. To recover the phase functions, a standard, Fourier-transform method, or parametric methods like a phase-locked loop (PLL) method, can be used. In the former case, the Fourier spectrum in the wavelength domain is computed and filtered out to obtain the reference spectrum and the overall phase information using a phase-unwrapping algorithm. In the latter case, the fringe phase deviations are traced dynamically in the independent variable domain, i.e. in the wavelength domain when the PLL method is applied to spectral interferometry. The PLL method was used to demodulate spectral fringes iteratively. A spectral fringe signal with a priori unknown carrier fringe frequency is considered and at the first iteration step a fringe phase equal to zero is supposed. The second iteration takes the demodulated phase found from the first iteration etc. As a result, the unwrapped phase function of the spectral fringes is found.

Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals

“…The algorithms can also be used to transform non-stationary interferograms into stationary ones. The stationary wavelength-domain channeled spectra are still appropriate to retrieve the spectral phase as a function of the wavelength [12,18].…”

“…For the spectrum recorded in the frequency (wavenumber) domain, these include a five-point [2,10] and a sevenpoint [7,9] algorithms, a Fourier-transform algorithm [5,8] and especially for the non-stationary channeled spectrum, a wavelet-transform algorithm [15]. For the spectrum recorded in the wavelength domain, the algorithms include a two-point algorithm [11], a Fourier-transform algorithm [12,13,18], a phaselocked loop method [12] and especially for the non-stationary channeled spectrum, the Kalman filtering method [14,21]. In the latter case, the spectral phase is reconstructed from the spectral interference signal, which is obtained from the recorded channeled spectrum subtracting the effect of the reference non-channeled spectrum.…”

Dispersion error of a beam splitter cube in white-light spectral interferometry

“…White-light spectral interferometry based on channelled spectrum detection and utilizing a broadband source in combination with a standard Michelson interferometer has been widely used for distance and displacement measurements [1][2][3], in optical profilometry [4][5][6][7] and for material characterization [8][9][10][11][12][13].…”

Dispersive white-light spectral interferometry with absolute phase retrieval to measure thin film

“…The spectral phase retrieval, which can be performed by various techniques like Fourier transform method [11], phase-locked loop method [11], and the Kalman filtering method [12], is crucial for absolute distance measurements in interferometers with thin films [2,5] characterized by the phase change on reflection that depends on wavelength and layer thickness. It has been shown for thin-film structures with one and two layers that the thicknesses of these layers can be determined with a high accuracy provided that the refractive indices are known [2].…”