1. Introduction

The exponentially increasing amount of fringe data generated by interferometers must be addressed. In multi-pulse train interferometers [1,2], the positions of two object mirrors are recorded in the envelope peaks of the interference fringes. A multi-pulse train interferometer comprises one balanced and one unbalanced Michelson interferometer. In the balanced Michelson interferometer, the pulse trains from a femtosecond optical frequency comb (FOFC) laser, reflected by the reference and object mirrors, are overlapped at a beam splitter. The two pulse trains interfere with each other, and the autocorrelation function between the pulse trains assumes its maximum value. The interference fringe detected by the photodiode (PD) reaches its maximum intensity. This corresponds to the position of the peak value of the envelope of the interference pattern. In the unbalanced Michelson interferometer, the different pulse trains reflected by the reference and object mirrors are overlapped at the beam splitter, where they interfere with each other. The cross-correlation between the pulse trains and the interference fringes assumes its maximum value, which corresponds to the position of the peak value of the interference pattern envelope. The spatial discrete coherence characteristics of the FOFC laser are utilized to record the position of the discrete absolute length. Finding the peak positions of the interference patterns is an important procedure for the multi-pulse train interferometer.

Estimation of the envelope of the interference pattern is based on communication theory, especially signal modulation and demodulation [3]. The position information of the object mirror is modulated (or recorded) by the carrier fringes. The Fourier transform converts the fringe intensity data into the frequency domain. Using the appropriate bandpass filter, we select the spectrum of the carrier frequency; using the inverse Fourier transform and appropriate signal processing, we demodulate the signal of interest. In the multi-pulse train interferometer, the position of the envelope peak of the interference fringes is of interest because this is where the two pulse trains overlap completely. This means that the distance between the reference and object mirrors is an integral multiple of the adjacent pulse repetition interval length.

Estimation of the envelope of the interference pattern usually involves the use of Fourier transform and inverse Fourier transform [3]. The Fourier transform is applicable to continuous functions. For discrete values (waveform data) obtained by sampling with an analog-to-digital converter, conversion from the time domain to the frequency domain is performed by the discrete Fourier transform (DFT). A fast Fourier transform (FFT) is a transform that enables quick calculation of the DFT; however, it is usable only for specific values of time and frequency spacing. As shown later, the envelope obtained by the DFT-based method has the same data length as the interference fringe signal. In other words, the entire envelope is regenerated. However, obtaining the complete envelope to acquire positional information regarding the envelope peak should not be necessary. In communication theory, we demodulate every signal, because every modulated signal contains its own information. For the multi-pulse train interferometer, the object of interest is the partial envelope around the envelope peak; therefore, reconstruction of the whole envelope of the interference fringes is unnecessary. Because our target is the position of the envelope peak, and not the whole envelope, partial envelope reconstruction allows smart signal processing.

The chirp Z-transform (CZT) was proposed in [4] for efficient DFT implementation over arbitrary frequency spacings. It has found applications in digital holography [5], spectrum analysis [6], optical processor design [7], chirp rate estimation [8], and medical diagnosis [9,10], among other uses. Further details on CZT can be obtained in [4,11,12]. In this study, a configuration utilizing CZT instead of DFT to reconstruct the partial envelope of the interference fringes to search for its peak position in a multi-pulse train interferometer is proposed for the first time. A real practical advantage of the partial reconstruction of the interference pattern envelope using CZT is the potential improvement in measurement resolution, as introduced in [4].

2. Principle

For convenience, we consider a Michelson-type interferometer, as shown in Fig. 1. A beam from the light source is collimated and divided into two beams by a beam splitter. The position of an object surface is fixed on an optical table and denoted by z_eq. The position of the reference surface, denoted by z_n, is modulated by a piezoelectric transducer (PZT) with time. Both beams are reflected by the object and reference mirrors and overlap at the beam splitter. By scanning the position of the reference mirror, interference fringes can be detected via a PD and a digital oscilloscope. The PD converts the detected optical signal to an electric signal, while the digital oscilloscope converts the obtained continuous electric signal into a discrete signal.

 

Fig. 1 Configuration of the Michelson-type interferometer.

Download Full Size | PDF

Let x(n), n = 0, 1, 2, …, L − 1, denote an L-length sample interference fringe.

The DFT-based envelope reconstruction approach, represented by E_DFT{}, is defined by the following equation, which governs the selection of the frequency spectrum of the interference signal in the frequency domain and the envelope reconstruction in the time domain of the time-discretized interference fringe x(n):

In the above equations, DFT{}, IDFT{}, and abs{} are the one-dimensional DFT, the one-dimensional discrete inverse Fourier transform (IDFT), and the modulus of the input value, respectively. The function BPF{} is the band-pass filter function [13] in the frequency domain.

The connection to the optical signal and the algorithm for processing it are as follows. DFT{x(n)} signifies the discrete Fourier transformation of the intensity data into the Fourier domain. The Fourier spectra of the interference fringes are symmetrical about f = 0 and separated by the frequency f_scan, which is the scanning frequency of the reference mirror. BPF{DFT{x(n)}} signifies the selective passing of the frequency components of the signal around the frequency f_scan; unwanted noise is filtered out by the band-pass filter. IDFT{BPF{DFT{x(n)}}} signifies the IDFT of the selected spectrum around f = f_scan into the time domain. Finally, the abs function takes the modulus to determine the interference fringe envelope. Because half of the frequency spectrum is neglected (namely f = −f_scan) due to the BPF function, the final value is multiplied by 2. Note that C_DFT(n) is data of length L. The whole envelope of the interference fringes is determined because a DFT-based estimator is used. Because the problem is defined as the reconstruction of the interference fringe envelope around its peaks, we can approach this from a different perspective: reconstructing a portion of the envelope that includes the interference fringe peak.

We now introduce our proposed algorithm. Because of the duality between the time and frequency domains of the Fourier transform (or Fourier inversion theorem) [14,15], we can reverse the order of the DFT and IDFT operators:

Using the CZT function instead of the DFT function, we define the CZT-based envelope reconstruction, represented by E_CZT{}, by the following equation:

Here, N is the number of points over which the CZT is calculated, a is the starting point and is defined as a = exp(j2πL_L/L), L_L is the calculation start point of the envelope reconstruction, and L is the number of samples. w is the set of points, starting from L_L, over which the CZT is calculated, given as w = exp(−j2π(L_H − L_L)/(NL)), where L_H is the calculation end point of the envelope reconstruction. In the case that N = L, w = exp(−j2π/N), a = 1, the result of the CZT-based method is equivalent to that of the DFT-based method.

This change is made because the CZT can be used to obtain the DFT of a finite-length sequence over a limited range of data, rather than over the whole range of the data. The envelope of interest is only the envelope near the envelope peak of a given signal, which is a fraction of the total envelope. Using the CZT-based method, the envelope data at any arbitrary point of the given signal data can be evaluated efficiently. Before magnifying (by CZT) the region of interest around the envelope peak, the approximate location of the region of interest should be identified. We can utilize prior knowledge on the envelope peak, which is the peak location where the signal strength is strong. Let I_max be the maximum intensity of the signal. We can only evaluate the envelope of the signal data whose maximum intensity is larger than 0.9 × I_max.

3. Experiment

The optical interferometer shown in Fig. 1 was constructed using a fiber laser light source identical to that in [16]. Further description is omitted here. To control the optical path difference (OPD) of the reference mirror, a compact one-axis positioning stage (PX 400, Piezosystem Jena GmbH) with a motion range up to 400 μm was used with a closed-loop resolution of 6 nm. The interference signal, detected by a fast sensitive InGaAs PD (2011-FC, New Focus, Inc.), was transferred to a digital oscilloscope. A plurality of the interference fringe data is stored in the digital oscilloscope. This interference fringe data is processed by a personal computer.

A description of the implementation of the CZT software is beyond the scope of this paper. The algorithm of the CZT was described in [17]. Although optimization of the CZT calculation has been reported in [18], because the computational complexity is low, the parameters have not been optimized in this proof-of-principle. Both methods (DFT and CZT) were implemented using the embedded functions in MATLAB software (MATLAB 8.1.0.604 (R2013a), The MathWorks, Inc.).

Here we show a qualitative comparison of envelope detection using the proposed method to that using the DFT-based method. These comparisons are a qualitative validation of the functionality of the new method proposed here.

The fringe under analysis and results of envelope detection using the two methods are shown in Fig. 2. The whole envelope detected using the DFT-based method, shown in Fig. 2(a), has the same data length as the interference fringe data. The partial envelope around the envelope peak using the proposed method is also shown in Fig. 2(a). For the proposed method, the specified parameters are the start and end points of envelope reconstruction INT[L/3] and INT[2 × L/3], respectively. The parameter L = 16384 pixels is the length of the sample interference fringe. The function INT[x] returns an integer from x. The partially regenerated envelope has one-third the number of data points of the interference fringe data. As evident in Fig. 2(b), which compares the two envelopes, if it is assumed that the DFT-method can reconstruct the envelope accurately, the envelopes are accurately reconstructed by both techniques. The slight difference between the two envelopes may result from truncation errors and/or round-off errors in the numerical calculations. Note that the operations INT[L/3] and INT[2 × L/3] introduce a small shift in the x-axis of the sample points of the CZT-based envelope relative to those of the DFT-based envelope. We performed the same signal processing for more than 20 different interference fringe data sets. All exhibited the same results. We note that the selected range of the partially reconstructed envelope can be further improved to reduce the number of data points requiring processing.

 

Fig. 2 Fringe processing: (a) interference fringe signal (blue line), reconstructed whole envelope using DFT-based method (black dashed line) and partially reconstructed envelope using CZT-based method (red points). (b) The difference between the two obtained envelopes. The red downward arrow indicates the position of the envelope peak.

Download Full Size | PDF

4. Conclusions

A new approach to partial envelope detection in multi-pulse train interferometers has been introduced. It is based on the Fourier inversion theorem and uses the CZT, rather than the DFT, for interference fringe processing. The CZT-based algorithm permits reproduction of the partial envelope and can be used for envelope peak detection. An example of the reproduction of the partial envelope, in comparison with the conventional DFT-based method, is presented here for the first time. The results show that the proposed CZT-based algorithm is reliable for partial envelope detection. In future works, we will exploit the rescaling functionality of the CZT for resolution enhancement by increasing the number of points embedded within the segment of the partially regenerated envelope. We also plan to apply the proposed method to envelope peak detection in vertical-scanning wideband interferometers.