## Abstract

In this paper we propose group refractive index measurement with a spectral interferometric set-up using a broadband supercontinuum generated in an air-silica Microstructured Optical Fibre (MOF) pumped with a picosecond pulsed microchip laser. This source authorizes high fringes visibility for dispersion measurements by Spectroscopic Analysis of White Light Interferograms (SAWLI). Phase calculation is assumed by a wavelet transform procedure combined with a curve fit of the recorded channelled spectrum intensity. This approach provides high resolution and absolute group refractive index measurements along one line of the sample by recording a single 2D spectral interferogram without mechanical scanning.

© 2006 Optical Society of America

## 1. Introduction

Spectral interferometry is a well-established optical measurement method in various application fields as profilometry [1], materials optical characterization [2] and optical imaging [3]. In particular, this technique is relevant for broadband optical communications applications. It is currently used for dispersion characterization of optical guiding media such as optical fibres [4–5]. In this case, polarization-mode dispersion, birefringence dispersion, or intermodal dispersion are measured.

Optical imaging systems used in the visible range are mainly limited by chromatic aberration due to optical component dispersion. Consequently, Spectroscopic Analysis of White Light Interferograms (SAWLI) which is known as a valuable tool for accurate dispersion measurements of transparent media receives increasing attention. High resolution measurements of phase refractive index over a wide spectrum range have been performed on prismatic samples or with a previous knowledge of the refractive index at one wavelength [6, 7]. Moreover, the analysis of group-delay dispersion by SAWLI is realized from a single shot image without any mechanical scanning when the tested sample thickness is known. By this way, the measurement is kept free from environmental noise and real-time measurement can be considered. Finally the main SAWLI’s limitation is fringes visibility, and solutions allowing its improvement are of considerable interest in analysis of complex media such as graded index optical components.

Dispersion measurements by spectral interferometry vary from one to another according to the kind of source [8]. Both thermal sources as tungsten halogen lamp or Super Luminescent Diode (SLD) are used. The first performs measurements over several hundred nanometers, but suffers from a low power density which limits SAWLI measurements on thin sample. On the contrary SLD, generally used for spectral-domain low-coherence interferometry, offers higher optical power density along a reduced spectral bandwidth; nevertheless, its higher temporal coherence limits spectral resolution and contributes to losses in spectral interferogram visibility.

In a recent paper dealing with profilometric measurements by SAWLI [1], we demonstrate that spectral fringes visibility is much improved by using a compact and efficient supercontinuum source [9] pumped by picoseconds pulses (at *λ*=1064 nm and *λ*=532 nm) and generated in a single mode air-silica Microstructured Optical Fibre (MOF). Indeed, the main advantage of this supercontinuum, rests on its spatial coherence, which induces high power density due to photon concentration in a single mode. However this source exhibits a strong spectral intensity peak at the doubled frequency (*λ*=532 nm) at the fiber output and generates noise all around this frequency. Since, a new generation of supercontinuum in a MOF is proposed [10] exhibiting a continuous, flat and homogeneous spectral intensity distribution, which is performed by using a passively Q-switch Nd:YAG laser delivering 600ps pulses at *λ*=1064 nm. With this new source, the spectral intensity peak at the doubled frequency does not appear in the interferometric signal leading to an improvement in the measurements presented in this paper.

In this paper, we validate by experimental results the SAWLI’s set-up combined with this new generation of supercontinuum source. We also present a phase calculation algorithm based on wavelet transform. By this way, group refractive indices are measured on samples in all the visible range (Δ*σ*=[1.25*µm*
^{-1}-1.87*µm*
^{-1}] corresponding to Δ*λ*=[530*nm*-800*nm*]) with high resolution from only one interferogram.

## 2. Experimental set-up

Figure 1 shows the experimental set-up [1] which associates the supercontinuum source with a Michelson spectral interferometer. A transparent sample is inserted in one arm of the interferometer. A set of two perpendicular cylindrical lenses L_{1} and L_{2}, combined with a filtering slit at the interferometer input allows to collimate the beam in the two perpendicular directions. Thus the sample analysis is performed along a vertical line. At the output of the interferometer, the spectral analysis is achieved on the horizontal direction by a 600 grooves/mm grating, a cylindrical lens L_{5} and a CCD camera of 484×782 pixels.

As a result, the vertical direction *y* on the detector is conjugated with the vertical direction on the inspected sample and the horizontal direction *x* corresponds to spectral information for each wavenumber σ of the analysed bandwidth, Δσ.

Then, at the output of the set-up, the light intensity distribution on *I*(*y*,*σ*), is expressed as a function of the wavenumber *σ* by the following equation:

where *I*
_{0}(*y*, *σ*) is the background intensity, *V*(*y*,*σ*) is the fringe visibility function and ΔΦ_{12}(*y*, *σ*) is the phase difference between the two beams:

where *n*(*y*,*σ*) is the refractive index, *e*(*y*) the thickness of the studied sample for the *y* height and *δ* is the optical path difference in air between the two interfering beams. Here, the considered sample is supposed to be homogeneous with equal thickness, thus *n*(*y*,*σ*)=*n*(*σ*) and *e*(*y*)=*e*.

## 3. Phase reconstruction procedure by wavelet transform

In order to perform absolute group refractive index determination, the phase calculation is achieved on a single interferogram obtained outside of the stationary phase region contrary to most of works [5, 11]. In this configuration the uncertainty on the phase calculation is reduced because the phase variation on the whole wavenumber spectrum is monotonous. This is possible thanks to the high spatial coherence of the supercontinuum source which authorizes a good visibility along a large wavenumber bandwidth Δ*σ* on the CCD detector (Fig. 2). The phase analysis procedure is now explained in this section.

The Eq. (2) shows non-linear variations of the phase with the wavenumber *σ* due to sample dispersion properties. For this particular case, it is advisable to use a simultaneous time-frequency analysis method [12]. For our application, the phase reconstruction is performed by using the Morlet wavelet transform which gives significant results in time-domain interferometry and overcomes the disadvantages of Fourier Transform [13].

More precisely, the experimental phase ΔΦ_{exp}(*σ*) is extracted from each y-line of the interferogram in two steps. The first step consists mainly in estimating the phase ΔΦ_{exp}(*σ*) term for each y-line by a Morlet wavelet transform treatment defined by Eq. (3)

where: *ψ*
_{a,b}(*t*) is known as the mother wavelet, *a* is the scaling factor related to the frequency *v*, and *b* is the pixel position on the analysed y-line interferogram.

This approach consists in scaling and translating a single template waveform on each interferogram line. The main purpose of this calculation is to find the curve described by the maximum values of |*W*(*a*, *b*)|^{2} on the a-b plane, called ridge. Then, the local frequency of the signal is deduced from the value of *a* at the pixel position *b* on the ridge. And the phase on the ridge is equal to the signal phase modulo 2*π*. This relative phase term is then expressed as a fifth order polynomial function in order to go through the following step.

The second step consists in precisely adjusting the parameters of the polynomial function by fitting the spectral intensity curve. These parameters are optimized by a least square method. The order of the polynomial function is previously chosen in order to obtain a good compromise between resolution and time consuming computing.

Finally, a fifth order polynomial function restores optical phase variations ΔΦ_{exp}(*σ*) over the whole analysed spectrum Δ*σ*. Relative phase ΔΦ_{exp}(*σ*) leads to the absolute first derivative phase
$\frac{\partial \Delta {\Phi}_{\mathrm{exp}}\left(\sigma \right)}{\partial \sigma}$
and therefore to the group refractive index *n _{g}*(

*σ*) of the inspected y-line sample according to Eq. (4):

Moreover the group-velocity dispersion (proportional to the first derivative of the group refractive index) could be deduced from the same measurement.

## 4. Experimental results

To start measurements, a 2.39 mm thick BK7 glass plate is set into one arm of the interferometer (Fig. 1). Then the reference mirror position corresponding to *δ* parameter is adjusted in order to observe reduced inter-spaced and resolved interference fringes as presented in section 3. The experimental interferograms are filtered in order to avoid artefacts during the numerical phase treatment (Fig. 2).

The phase calculation is applied on each interferogram line by a wavelet transform algorithm performed by ridge extraction (Fig. 3).

This algorithm leads to the experimental absolute group refractive index law along the whole 18mm line of the BK7 glass plate [Eq. (5) and Fig. 4(a)]. The maximum standard deviation from the mean value reported on the 18mm line for group refractive index over the whole analysed wavenumber bandwidth [Fig. 4(a)] is 1.2×10^{-4}.

The experimental group refractive index law for one y-position is compared to the model deduced from the Sellmeier equation [Fig. 4(b)]. The maximum error between the Sellmeier model and the experimental values reaches 3.2×10^{-4} over the whole analysed wavenumber bandwidth.

The phase reconstruction gives then a very weak error on the group refractive index. This result validates the Morlet wavelet algorithm.

The repeatability of the proposed method has also been verified by applying the same process for three different mirror positions 10µm apart from each others (*δ*
_{1}=-1210*µm*, *δ*
_{2}=-1220*µm* and *δ*
_{3}=-1230*µm*). The refractive index laws deduced from these three interferograms are represented in Fig. 5 and the maximum error between the three experimental group refractive index profiles is 3.5×10^{-4}. This result validates the repeatability of the proposed method.

## 5. Conclusion

In conclusion, the ability to achieve a one line absolute high resolution group refractive index profile over a wide spectral bandwidth from the analysis of a single shot image without mechanical scanning is experimentally demonstrated. The originality of this work consists in the use of a supercontinuum source, which authorizes the observation of interference fringes pattern outside of the stationary phase region and in a wavelet transform algorithm phase calculation. Phase calculation using wavelet transform algorithms reaches a 3.5×10^{-4} calculated resolution value on group refractive index measurements and could lead to the group-velocity dispersion calculation. It is also possible to measure the refractive index law with the same resolution by knowing its value for one wavenumber. Finally the resolution of this method authorizes inhomogeneous sample characterization such as graded index optical components.

## Acknowledgment

The supercontinuum source was provided by Xlim team and this work was supported by “STICOPTO Massif Central”.

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