1. Introduction

Refractive index is a dimensionless scalar that defines the optical properties of a given medium and for this reason its knowledge is of paramount importance in optical design and for the understanding of light propagation. It is fundamentally linked to the electronic structure of their constituents and can be modified by external parameters such as temperature, electric and magnetic fields, pressure, and others. This gives rise to a variety of optical phenomena such as chromatic dispersion, optical Kerr and Faraday effects, photoelasticity, and so on, motivating the development of new techniques aimed at the accurate determination of the refractive index and how it depends on those external parameters. Refractometry of transparent optical samples is currently accomplished by a number of well-established techniques, most of them based either on the Snell-Descartes law or in interferometry. However, it is not unusual to find drawbacks in these techniques and one of them is the difficulty in accommodating any bulky apparatus to change external parameters such as temperature or pressure. In this context, the present work proposes a technique based on spectral-domain low-coherence interferometry to circumvent this difficulty.

Spectral-domain interferometry with low-coherence sources has attracted great interest for its usefulness in characterizing reflections from closely spaced surfaces in optical systems. This feature allowed the development of important applications in optical engineering and biomedical fields, such as the optical coherence tomography (OCT) [1-4] and the simultaneous measurement of the geometrical thickness and group index of transparent samples [5-10]. Although the OCT technique started with the usual dual-arm Michelson interferometer, it evolved to a common-path interferometer, which is more compact and stable [3,4]. On the other hand, the measurement of thickness and group refractive index of transparent samples has a number of variants, most using the dual-arm low-coherence Michelson interferometer [5-7]. Techniques using a combination of Michelson and Fabry-Pérot (FP) interferometers [8], modified OCT [9] and low-coherence interferometry with confocal optics [10] were also introduced, among others. The later has the advantage of measuring, besides the geometrical thickness, both the phase (n_p) and group (n_g) refractive indices that are related according to: n_g = n_p - λdn/dλ. Owing to the material dispersion, these values can differ significantly. Each of above mentioned methods has its own advantage but most of them have the drawback of needing to move some optical component. For instance, the white-light spectral interferometry of [7] relies on the accurate displacement of the reference mirror of an uncompensated Michelson interferometer while in those techniques using a FP interferometer or the modified OCT, it is necessary the precise rotation of the sample [8,9]. Although the dual-arm white-light Michelson interferometer has no moving parts [5,6], the sample needs to stay close to a reference surface and this inhibits the use of any bulky temperature apparatus such as an oven or cryostat. Furthermore, two measurements, with and without the sample, are needed.

The present work reports on a simple low-coherence interferometer where two beams propagate parallel to each other in the same arm, with just one of them passing through the sample. Besides the interference between the two parallel beams, the sample behaves as a low finesse FP etalon, providing important complementary information. The device is compact and stable, has no moving parts, needs just one measurement and the sample can be located in any position along the optical path, allowing the introduction of an apparatus to change external parameters, such as an oven to determine thermooptic and expansion coefficients.

2. Experimental section

The schematic diagram of the experimental setup is shown in Fig. 1. The light from a super luminescent emitting diode (SLED) centered at 840 nm is launched into a 3 dB optic coupler where just one of the output arms is used. This coupler has a role different from that of a beam splitter usually found in two-beam interferometers such as Michelson or Mach-Zehnder, but similar to the one used in the common-path interferometer. In the first, the beam-splitter is necessary to recombine the beams in order to produce interference, while in the later its only purpose is to direct part of the light into a spectrometer after the interference has already taken place. A 1x2 coupler would be more adequate, but the setup was initially planned for a conventional two-path Michelson interferometer and thus a 2x2 coupler was used. As mentioned, part of the light reflected from the measurement arm is sent to a portable spectrometer with a resolution of about 100 pm and the spectrum acquired is Fourier transformed to yield the relevant optical path differences as usually done in spectral-domain interferometry.

 

Fig. 1 (a) Schematic diagram of the one-arm interferometer used for the simultaneous measurement of thickness and group index of a plane-parallel transparent plate. Here, SLED is the super luminescent emitting diode, PS is the portable spectrometer, S is the sample and RF is a reference flat. (b) Expanded view of the measurement path.

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The beam coming from the output port is collimated to a diameter of about 4 mm and sent in the direction of a glass reference flat, RF, which plays the role of a mirror. We couldn’t use a plane aluminum mirror because the returning light saturates the spectrometer CCD even for the smallest integration time available. The upper part (I) of this incoming beam goes directly to the back mirror while the lower portion (II) passes through the plane-parallel sample. Although there is just one incoming beam, the two halves propagate through different paths and thus the device can be considered as a two-beam interferometer. The border of the sample has to be well defined to avoid losses of the incoming beam. Owing to the spectrometer resolution (δλ ≈0.1 nm), the maximum sample optical path, given by [11]:

The method is based on the fact that the parallel plate plays a double role - it acts as a low finesse FP etalon for beam (II) and also provides an optical path difference between the two parts of the parallel beam.

The intensity of beam (2) reflected from the plate is given by [12]:

After returning from the back reflector, parallels beams (I) and (II) are focused into the single mode fiber and produce a two-beam interference, with an optical path difference of:

This single arm interferometer is easy to align and, when compared to a two-arm interferometer, does not need any compensating adjustment in one arm when the position of an optical component in the other arm is moved. The shared single arm also increases the interferometer’s stability and reduces its sensitivity to vibrations because the beam overlap is automatically maintained. Furthermore, the sample can be placed in any position along the optical path and there is no need of removing it during the measurement, as done in [5] and [6].

3. Results and discussion

In order to test the method, measurements were carried out in Herasil 102 fused silica, Schott B270 Superwhite crown glass and borosilicate cover glass, whose physical thicknesses were initially measured with a precision digital micrometer gauge (1 μm resolution). Figure 2 shows the results achieved for fused silica with a thickness of 1.198 mm. As seen in (a), the SLED used has a non-Gaussian spectral profile but this does not affect the results, just increases the signal close to the origin. The Fourier transformed signal in (b) presents several peaks that are discussed next. FE corresponds to the interference arising from reflections at the fiber end facets, observed even when the collimated beam is blocked before the sample. These reflections can be used as self-reference in a dual-arm Michelson interferometer, as done in [4], but in our case we don’t find it necessary. If a 1x2 coupler were used, such interference would not be observed. The relevant parameters measured, Δl and Δ, were found to be 1.76064 and 0.5603, which result in t = 1.200 mm and n = 1.4668 ± 0.0015. There is another well-defined peak around 0.2802 mm that we attribute to the fact that the beam is not perfectly collimated such that part of the incoming light passes through the sample but returns outside, or vice-versa, yielding exactly half of the optical difference, Δ/2. In order to corroborate this hypothesis we carried out measurements where the sample was located closer to the reference flat and verified that this peak decreases. This occurs because the beam does not propagate enough to miss the sample. In the case where the sample is just a few millimetres from the reference flat, this peak completely disappears.

 

Fig. 2 (a) Spectrum of fused silica obtained with the single arm interferometer. (b) Optical path difference obtained from the spectrum shown in (a).

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The geometrical thickness value was just 2 μm thicker than the value found with the micrometer gauge. It is known that the depth resolution of the interferometer, δz, is limited by the coherence length of the light source, l_c according to [13]:

Figure 3 shows similar results obtained with a Schott B270 Superwhite crown glass slab, whose thickness was mechanically measured as 0.957 mm and whose group refractive index was reported as 1.5321 ± 0.001 at 814 nm and room temperature [10]. After Fourier transforming the spectrum of Fig. 3(a), the values Δl = 1.46224 and Δ = 0.50732 were obtained, resulting in t = 0.955 and n_g = 1.5312, which is also in good agreement with that found in [10]. The peak at Δ/2 was also observed in this case.

 

Fig. 3 Spectrum of a B270 sample (a) and its Fourier transform showing the corresponding optical path differences (b).

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Finally, Fig. 4 shows the results obtained with a Schott D263M borosilicate cover glass, whose thickness was measured with the micrometer gauge to be 0.148 mm and whose group index was determined at 814 nm as 1.5548 [10]. Owing to the small thickness of the sample, the values of Δl and Δ are near the origin and are therefore more likely to be influenced by low frequency noise. Indeed, this can be observed in Fig. 4(b), but even though, the values of Δl, Δ and Δ/2 can be easily determined. These values were found to be Δl = 0.22739 and Δ = 0.0804, resulting in t = 0.147 and n_g = 1.5468.

 

Fig. 4 (a) Spectrum of a borosilicate cover glass obtained with the single arm interferometer. (b) Optical path difference obtained by Fourier transforming the spectrum shown in (a).

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To summarize the results obtained for the samples studied, Table 1 presents the values measured in comparison with the mechanically determined geometrical thickness and values of refractive index found in [10]. From these results we conclude that the thickness resolution provided by the one-arm low-coherence interferometer is at least 2 μm. It is worth to mention that the optical measurement is performed right at the border of the sample which may not be so flat. Concerning to the group index, we compared the results to those of [10] which also has errors. The results agree within 0.1%, except for the cover glass sample. In this case, owing to small thickness, Δl and t are small numbers and their ratio has an intrinsically higher error.

Table 1. Refractive group index and physical thickness of the samples studied at 840 nm.

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4. Conclusions

In summary, we proposed a one-arm low-coherence interferometer that allows the measurement of the geometrical thickness and group index of plane-parallel optically transparent samples. There is only one incoming beam propagating through the single arm, but it can be considered as two-beams because about half of it passes through the sample and the other half does not. The beams propagate through a nearly common path and this makes the device very compact and stable, almost immune to mechanical vibrations. Another advantage of sharing nearly the same optical path is that any air drift is automatically compensated. Furthermore, the optical device has no moving parts, needs just one measurement and the sample can be placed in any position along the optical path. This allows the introduction of an oven, for instance, that permits the investigation of thermooptic and expansion coefficients of new glassy materials.