1. Introduction

White light interferometries (WLIs) [1, 2] based on the use of a broadband source in combination with a standard Michelson or a Mach-Zehnder interferometer is widely used in various fields such as material characterizations, optical communications, profilometries, and displacement measurements. The WLI usually utilizes a temporal method [1] or a spectral method [3]. The temporal interferometry employs a scanner to sweep the wavelength over a spatial distance with a constant speed. Alternatively, the spectral interferometry is realized when the two arms of the interferometer are nearly equalized for a certain wavelength. Both methods calculate chromatic dispersions from double differentiations of the relative phase with respect to the optical angular frequency. However, in experimental conditions, there are always noises on the retrieved phase, such that the numerical differentiation on retrieved phase could result in huge numerical errors, especially for the second order differentiation. For transparent materials, polynomial or Sellmeier functions are usually used to fit the relative phases because their group delay (GD) and group delay dispersion (GDD) are regular. Nevertheless, for ultra-broadband chirped mirrors, since their GD and GDD are usually associated with large irregular oscillations, curve fittings cannot be applied with polynomial or any other regular functions. Therefore, to accurately evaluate the chromatic dispersions characteristics, a direct reading of the high order phase, GD or GDD, rather than the relative phase, is desirable.

We have proposed a straightforward and robust technique, joint time-frequency analysis with wavelet-transform technique, for direct GD extraction [4]. We demonstrated that either monotonic or complicated GD could be easily extracted and predicted the technique had potential values for GD and GDD extractions of chirped mirrors [4].

In this paper, we measured white light spectral interferograms of a chirped mirror for different matched wavelengths, and extracted GDs and GDDs from the measured interferograms with both the proposed method and the conventional phase differentiations method. We further investigate the accuracy dependence on the alignment of the interferometer, and show that the accuracy less depends on the balanced path length of the two arms of the interferometer in comparison with the conventional method.

2. Group delay extraction from derivative of phase

We employed a standard Michelson interferometer for recording the interferograms. The light source was a metal halogen lamp coupled to a single mode optical fiber. We used a lens to collimate the beam from the fiber. A home made chirped mirror was mounted on one arm of the interferometer and a silver mirror on the other arm to reflect the beam. The interferograms were recorded by a spectrometer (Ocean Optics, HR4000).

The optical length difference of the two arms in an interferometer governs the fringe spacing, of which the largest spacing corresponds to a wavelength that has zero path length difference. A gradual reduction of the fringe spacing is seen towards both sides of the balanced wavelength. Figure 1 shows the spectral interferograms taken on both sides of the balanced central wavelength. The interferograms in the left column were taken on the short-wavelength side of the balanced wavelength (>600 THz or <500 nm) and those in the right were taken on the opposite side (<300 THz or >1000 nm).

 

Fig. 1. White light spectral interferograms of the chirped mirror for different matched wavelengths.

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Conventional GD extractions are from the derivative of the relative phase [1–3, 5]. First, one extracted the spectral phase of the interferogram with a Fourier transform [1–3] or a wavelet-transform [5], and then obtained the GD and GDD by single or double differentiations of the extracted spectral phase. The GD and GDD are the derivative and the quadratic derivative of spectral phase with respect to the optical angular frequency, respectively. In the procedure of spectral phase extraction with a Fourier transform, the filter should be carefully selected: it should be broad enough to include the GD bandwidth, and as narrow as possible to exclude the DC and noises. In the procedure of spectral phase extraction with wavelet transform, the phase is read from the ridge and no filter is applied [5, 6].

We have analyzed the GD directly read with the proposed technique, and the GD obtained with differentiation of phase retrieved with wavelet-transform technique [4]. The results illuminated that GD retrievals from phase differentiation and phase extraction were redundant to-and-fro processes and prone to huge error, in comparison with the direct reading of GD [4]. In this section, we retrieve GD from a Fourier transform, and demonstrate the direct reading of GD also has advantages over conventional Fourier transform technique.

We first transformed the spectral interferograms, and selected “suitable” filters to extract the phases. Then the GDs were obtained from the differentiation of spectral phases with respect to the angular frequency [Fig. 2].

 

Fig. 2. Extracted GDs of the spectral interferograms from the derivative of phases with Fourier-transform. GD: group delay.

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Figure 2 showed the GDs obtained from the conventional differentiation of phases extracted from a Fourier transform technique. It can be seen that all the GDs had oscillation structures in most of the spectral interference range, which mainly came from the noise of the phase differentiation. For the spectral interferograms measured at matched wavelengths larger than 600 THz, the extracted GDs could agree relatively well; however, for the matched wavelengths less than 300 THz, the GDs contained too much noise to be consistent with each others.

We then extracted the GDDs by taking another differentiation of the GDs, i.e., by taking double differentiations of the spectral phases. The extracted GDDs were shown in Fig. 3.

Figure 3 displayed the extracted GDDs from double differentiations of the phases. In Fig. 3, the GDDs derived from conventional phase differentiations showed such dramatic differences and contained so much noise that it was difficult to determine the GDD of the chirped mirrors. Figure 3 demonstrated that the conventional GDD extraction technique with phase differentiations could produce huge error. It also hinted that the procedure of differentiating amplified the noise of phases; therefore, direct reading GD to reduce error was anticipated.

 

Fig. 3. Extracted GDDs from the double differentiations of phases extracted from a Fourier-transform technique. GDD: group delay dispersion.

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3. Group delay direct extraction from ridge of wavelet-transform

We have demonstrated group delay can be directly extracted from the ridge of wavelet-transform [4]. By defining t(ω) as the local period of the interferogram for a particular frequency ω, the relative spectral phase φ(ω) can be written as:

where ω is the optical angular frequency.

GD is the derivative of spectral phase φ(ω) with respect to optical angular frequency ω ; therefore:

Equation (2) shows that the local period of the interferogram is exactly the group delay. Wavelet-trans form is a powerful tool for joint time-frequency analysis [4–7]. The wavelet-transform of a spectral interferogram f(ω) is expressed as [5]:

where ψ(ω) is a mother wavelet function and $\psi \left[\frac{\left(\omega -\omega \prime \right)·t}{2\pi }\right]$ is her daughter wavelet functions. Herein ω′ is shift factor, and 2π/t is dilation factor.

Wavelet-transform is the comparison between spectral interferogram and daughter wavelets, which reflects the similarity between the interferogram and daughter wavelet functions. The amplitude of wavelet-transform reflects the degree of their similarity. When the period of a daughter wavelet is the same as or the closest to that of the local spectral interferogram, the amplitude of wavelet-transform reaches the maximum value at that local frequency position. Therefore, the frequency of daughter wavelet at the ridge [7] is the local frequency of the interferogram, i.e., the position of the ridge in the time-frequency plane is exactly the GD curve with respect to the angular frequency.

We applied wavelet-transform on the six spectral interferograms shown in Fig. 1. The analysis wavelet is Gabor wavelet $\left[\psi \left(\omega \right)={\left({\sigma }^2\pi \right)}^{-1/4}·\exp \left(-\frac{{\omega }^2}{2{\sigma }^2}+\mathrm{i\eta \omega }\right)\right]$ [7] with (σ = (2ln2)^-1/2 and η = 1 [4, 6, 7]. The wavelet transforms for the six spectral interferograms were shown in Fig. 4.

 

Fig. 4. Wavelet transforms of the white light spectral interferograms.

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Wavelet-transform displayed the interferograms on a two-dimensional plane, on which one axis is time and the other is frequency [6]. The local periods of the interferograms reflected on the positions of the ridges [7], i.e., the maximum values along time axis at each frequency point. We detected the ridges (GDs) of the spectral interferograms and showed them in Fig. 5.

 

Fig. 5. Extracted GDs from the ridges of wavelet-transform for the spectral interferograms.

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It can be clearly seen that in Fig. 5 all the relative GDs extracted from the spectral interferograms agreed excellently with each other, for either the matched wavelength points less than 300 THz or those beyond 600 THz. Further comparisons were performed by superimposing them together and showed the differences fluctuated within ±4 fs for all the six GD curves in the entire spectral interference range (from 300 THz to 600 THz). All these proved wavelet-transform was a robust technique for GD extractions. It also demonstrated that the technique had a tolerance for the optical length of the two arms of the WLI, which greatly saved effort on the alignment of the interferometer.

By taking a single differentiation of the GDs, we obtained the GDDs [Fig. 6]. Figure 6 showed there was a good match between the extracted GDDs despite that those interferograms were recorded for different imbalances adjusted in the interferometer, especially in the spectral range from 400 THz to 550 THz. Beyond this range there were slight differences; however, the modulation periods of the GDDs could match well.

 

Fig. 6. Extracted GDDs from the ridges of wavelet-transform for the spectral interferograms.

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Figure 6 also demonstrated that differentiation was a procedure prone to introducing noise. The conventional GD extraction took one differentiation of the phase, which introduced noise into the extracted GD; however, direct reading of GD applied no differentiation, therefore the GD has a high precision. In the same way, the conventional GDD extraction took double differentiations, which made noise more prominent; however, direct reading of GD cut differentiation to once, which greatly reduced noise.

4. Conclusions

We have analyzed the GD and GDD extraction from different matched wavelengths of spectral interferograms with conventional Fourier-transform and the wavelet-transform technique. The results showed that the GD and GDD obtained from direct reading matched more excellently for all the measured interferograms and in a broader range of spectral interference, in comparison with conventional Fourier-transform technique. This technique does not require a critical alignment of the WLI, and can save the effort on alignment. The precision, flexibility and robustness of this technique would not only contribute to characterizations of ultra-broadband chirped mirrors, but also benefit various applications of white light spectral interferometer such as material characterizations, optical communications, profilometries, displacement and thickness measurements.