## Abstract

We present a characterization technique of wide-area sub-wavelength structures. The optical bench is based on lateral shearing interferometry, which allows an accurate complex transmittance (phase and amplitude) measurement. The experimental validation is made in the long-wavelength infrared domain; more precisely we work in the integrated 8–9 µm spectral range. Measurements of the transmitted amplitude and phase shift reveal a good agreement with respectively experimental results based on Fourier Transform infrared spectrometry, and theoretical simulations.

© 2008 Optical Society of America

## 1. Introduction

Miniaturizing optical functions such as light polarization, light confinement, spectral filtering, or others, in the perspective of developing a panel of integrated optics is a soaring requirement. From this point of view, surface plasmons engineering [1] is becoming an efficient tool which allows a wide variety of applications, such as the coding of complex (amplitude and phase) transmittance patterns on wide [2] or narrow spectral bands [3, 4]. The realization of such devices benefits from the high maturity of the microelectronics fabrication techniques, which makes it possible to structure the matter at the scale of the light wavelength in the visible or in the infrared domain.

A scanning electron microscope (SEM) characterization of these structures can only lead to their geometrical parameters: it reveals useful as a quality control of the deposits thickness or the period of the micro-structures. In addition to this preliminary control, there appears a need for the precise optical characterization of sub-wavelength structured materials (or so-called metamaterials), for mainly two reasons. The first motivation is that metamaterials induce spectral or spatial resonance effects, which are highly dependent on the various geometrical parameters (period if any, hole or slit form and size⋯) and on the intrinsic quality (uniformity, roughness) of the device that has been processed. By measuring this optical *signature*, one can thus deduce useful parameters on the device.

The second motivation is that the use of metamaterials paves the way to optical index engineering [5]. Indeed, within some limitations, metamaterials can be treated as homogeneous media [6, 7] described with *effective* optical properties. But then, there is a need for the determination of the complex (modulus and phase) coefficients of the effective permittivity and permeability tensors, for instance through the measurement of the metamaterials signature (transmission, reflection or absorption) all over the optical and angular spectra.

Various techniques have been proposed to determine the phase shift induced by the sub-wavelength gratings (use of a Michelson interferometry [8], use of walk-off or polarization interferometry [9], or even the analytic inference of the phase from a transmission measurement [10]) or to evaluate the spectral transmission of the device (photo-spectrometer or Fourier Transform Infrared spectrometer). It is noteworthy that none of the described techniques allows a spatially resolved evaluation, or simultaneous phase and amplitude transmission measurement, which are key features for the effective index determination or for the quality control of a wide area device.

Lateral shearing interferometry (LSI) allows to simultaneously determine the amplitude and the phase [11] generated by wide area optical elements. We thus propose a LSI-based characterization bench for sub-wavelength structured materials. A first demonstration has been performed in the particular case of normal incidence and for a narrow spectral band around 8 µm, but the generalization to the whole spectro-angular spectrum is straightforward.

The paper is organised as follows: in the first section we present the test sample made of sub-wavelength gratings. The second section is dedicated to the basics of the lateral shearing interferometry, and we present eventually the experimental results of the characterization.

## 2. Sub-wavelength gratings

#### 2.1. Design of the gratings

We make use of hybrid dielectric/metallic sub-wavelength gratings to code a transmittance pattern. Namely, a given level of transmittance is obtained using a sub-wavelength grating with a dedicated geometry. One may vary the metal layer thickness *h*, the width *w* or the period *d* of the slits. Additionnal anti-reflection layers are also used in order to reach the highest levels of transmission (Fig. 1).

The sub-wavelength metallic gratings are designed using a quasi-analytical one-mode model widely studied in the literature [12, 13]. It allows a fast study of metallic gratings with a wide range of geometrical parameters (metal layer thickness *h*, slit width *w* and period *d*), and provides a simple understanding of physical mechanisms involved in the transmission process. This model is valid as far as the metal thickness *h* and the slits widths *w* fulfill the following conditions: (i) each slit behaves as a one-mode metallic planar waveguide, *ie* the only guided mode is the TM0 mode (*w*<*λ*/2), (ii) other (evanescent) modes in the slits are sufficiently attenuated and can be neglected (*h*>*w*/2). As a consequence, the whole structure transmits only TM (transverse magnetic, *ie* magnetic field parallel to the grating slits) polarized waves, whereas TE (transverse electric) waves are totally back-reflected.

#### 2.2. Structure of the sample

The basic structure is made of a gold grating deposited on a gallium arsenide (GaAs) substrate, which is first double-side anti-reflection (AR) coated using silicon nitride (Si_{3}N_{4}) (Fig. 1). The period *d* and the gold thickness *h* are chosen in order to avoid the apparition of diffracted waves in the air and in the substrate and to fulfill conditions (i) and (ii): *d*=2 µm and *h*=830 nm. Then, the transmitted amplitude is simply a function of the slit width *w* [14].

The demonstrator is made of six domains (Fig. 2), each coding a distinct intensity level, and corresponding to specific geometrical parameters (*w _{n}*) reported on Table 1. The geometrical extension of each domain is 2×2 mm, which allows an easy experimental study. The technological realization process of the demonstrator is described in [14]. The structure of the demonstrator as well as a SEM photograph of the sample are shown on Fig. 2.

From Table 1, it may be pointed out that the use of metallic gratings induces an additional variation of phase, which is noted *φ _{n}*, so that the effective transmission of each grating writes Γ=

*T*·exp(

_{n}*iφ*). For illustration, a simulation based on the one-mode model is proposed on Fig. 3. It shows a device made of five sub-wavelength metallic grating domains of different slit widths

_{n}*w*and one domain corresponding to the pure AR-coated layer; this device diffracts a plane wave under normal incidence. Only the transmitted waves are taken into account in the diffraction pattern. One can notice that the transmitted plane wave is split into six sub-waves, each being slightly phase-shifted with regard to the others. The phase shifts

*φ*, are indicated in Table 1.

_{n}This additional phase shift has already been pointed out [14], but to our knowledge, it has never been measured. As described in the next section, we propose a solution based on the optical analysis using a lateral shearing interferometer.

## 3. Quadri-wave lateral shearing interferometer

Lateral shearing interferometer (LSI) based on a diffraction grating is a well-known wave front sensor [11], successfully used for classical optical testing, laser beam analysis or turbulence analysis for adaptive optics loops. The fundamental principle is to make two replicas of the impinging wave front under study interfere. As the replicas are coherently superposed, this technique does not require a reference wave; this is of great interest in term of simplicity and compactness of the measurement device. Recently a new class of lateral shearing interferometers has been proposed: the principle is strictly identical to classical LSIs except that they rest on the interference of more than two replicas of the impinging wave front [15, 16, 17]. For instance, quadri-wave lateral shearing interferometer (QWLSI) makes four replicas of the wave front interfere. The division of the incident wave into the four interfering replicas is operated by a diffraction grating called a Modified Hartmann Mask (MHM) [18]. This particular device allows the full spatial characterization (except polarization) of a wave front in a single measurement, *ie* the amplitude and phase transverse (*x*,*y*) measurement of the electric field.

The interference pattern is made of bright spots and undergoes deformations, that is shifts of the spots, when an aberrated wave front impinges on the QWLSI. As in the case of the classical two-wave LSI, the analysis of the shift of the spots easily leads to the derivatives or the difference quotient of the wave front [19, 20]. This demodulation is preferentially done by a Fourier analysis [21, 15]. In the case of a QWLSI, we obtain bi-dimensional map of derivatives.

In the particular case of the six-domain sample described before, the phase to be measured is discontinuous. Nevertheless, it has been experimentally shown that LSIs, and QWLSIs in particular, allow to measure such a segmented wave front [17, 22]. As a diffraction-based LSI makes two tilted and shifted replicas interfere, the interferogram is sensitive to the difference between the duplicated phases. For the sake of simplicity, let us consider a two-wave LSI, whose diffraction grating diffracts only two tilted replicas of the impinging phase *φ*(*x*,*y*). The interferogram is then made of fringes whose deformations are linked to the difference Δ* _{φ}* (

*x*,

*y*):

where *s* is the lateral shearing distance along the *x*-direction. In the case of a continuous phase, this expression can be approximated by the derivative, thanks to a first order approximation:

this shows that the interferogram deformations are directly linked to the derivative of the impinging phase, as it was asserted before. However for a discontinuous phase, this approximation is not valid anymore on the discontinuity. If we consider the case of a perfect segmented phase made of two constant phases separated by a *h*-high step, the phase difference Δ_{φ}(*x*,*y*) is made of a *s*-wide, *h*-high crenel (Fig. 4). In this case, the interferogram allows to measure the step height directly, without reconstructing the incident phase. However it is worth noting that the same reconstruction algorithm (for example with iterative Fourier treatment [23, 24, 15]) can be used for continuous or discontinuous phases. In the case of the reconstruction of a segmented phase, it behaves as if the *h*-high crenel was an approximation of the derivative of the phase. This results in a slight smoothing of the edges.

The same reasoning can be followed with a QWLSI. The phase difference is then computed along the *x*,*y*-directions, as far as the steps are oriented in the same way as the MHM.

The transmitted amplitude can be determined by the analysis of the 0th-order. This is equivalent to a simple descreen of the interferogram.

The Fig. 5 summarizes the data extraction from the initial interferogram (Fig. 5(a)). The spectrum (Fig. 5(b)) is computed and two harmonics are used to compute two derivatives along the *x*- and *y*-axis (Figs. 5(c) and (e)). These two derivatives can be integrated to reconstruct the phase (Fig. 5(f)). On the other hand, the 0th-order determine the transmitted amplitude (Fig. 5(d)).

## 4. Experimental validation

#### 4.1. Setup

The diffraction grating based QWLSI offers the crucial advantage to be easily transposed in the infrared domain. The sub-wavelength gratings of the tested sample (see section 2.2) are optimized for the 5–9 µmspectral domain. Thus we use a black body at *T*=1200°C as source and an uncooled microbolometer array (320×240 pixels) as detector. The infrared radiation is collimated by a catadioptric collimator (focal length: 760 mm; diameter: 120mm) so that the incident wave is plane (see Fig. 6)

A polarizer can be added on the optical path in order to select the polarization which is transmitted by the sub-wavelength gratings. This precaution ensures that only TM modes are transmitted. However, the parasitic (high order) TE-modes transmitted by the sub-wavelength gratings have been measured and are very weak (less than 1% in the [5 µm,12 µm] spectral domain). The use of a polarizer is thus optional.

The lateral shearing distance *s* is determined by the distance between the detector and the diffraction MHM grating. This distance has to be chosen so that the lateral shearing distance is about a few bright spots on the detector in order to be correctly resolved. Typically, for a pixel size of 35µm and a MHM period of 400µm this means a distance greater than 10mm to have a shearing distance of four spots.

#### 4.2. Experimental results

The recorded interferogram (Fig. 7(a)) is used to compute both derivatives (Fig. 7(b) and (c)) along the *x*- and *y*-axis and the transmitted amplitude (Fig. 8(a)). From these two derivatives, the reconstruction algorithm leads to a phase pictured on Fig. 8(b). A reference measure has been carried out so as to suppress the main aberrations produced by the collimator or the variation in the MHM thickness or flatness; this simply consists in an acquisition of the interferogram without the sample under test. However residual slowly-varying aberrations remain, which are linked to the GaAs substrate of the sub-wavelength structured sample. This could be improved *a posteriori* by filtering the low-order aberrations by a projection on the Legendre’s polynomials, for example.

The computed derivatives are used to determine the step height. Theoretically, as shown before, we should then measure a crenel whose height is the phase difference induced by the sub-wavelength grating. Each grating is imaged onto areas of around 50×50 pixels. By averaging the measures along the edges of the steps (*ie* along the *x*- or *y*-direction) one may improve the signal-to-noise ratio and thus the accuracy of the experiment.

#### 4.3. Comparison with expected results

The measured values for the phase shifts induced by the sub-wavelength gratings are summarized in Table 2. The experimental phases are an average in the spectral band of the study, that is 8–12 µm. Since the phase variations on this spectral range are negligible (less than a few percent), the simulated phases are given at 8µm only. One obtains a good agreement between the measured values and the theoretical ones.

The uncertainties given in the table are evaluated thanks to the ergodicity principle: the phase shift is measured around 30 times along the domain. There appears to be a discrepancy between the experimental and theoretical values for the fifth grating (slit width of 1460 nm). A possible explanation may be found in an eventual non uniformity of the sample surface (thickness of the substrate or thickness of the AR coating): indeed, a phase shift of 0.1 rad is equivalent to a thickness defect of *λ*/60, *ie* of a few tens of nm in this case.

The transmission of the substrate and the sub-wavelength gratings measured by the QWLSI provides the average value on the 8–12µm spectral band. For comparison the transmissions are evaluated with a calibrated Fourier Transform IR (FTIR) spectrometer (Bruker Equinox55) and averaged on the same spectral band. These values are our reference for the measure by the QWLSI. The QWLSI evaluation of transmission are then based on the response linearity of the microbolometer: thanks to two values of known transmissions, the transmission of the sub-wavelength gratings can be evaluated. We have carried out the calibration by matching the measured transmission of both the first grating and the bare substrate and the gray-levels given

by the microbolometer. The results are reported on Fig. 9. Once again, the agreement validates the accuracy of the QWLSI measurement.

## 5. Conclusion

We made use of lateral shearing interferometry for the optical characterization of wide-area, sub-wavelength structured devices. For this, we first proposed a sub-wavelength structured sample, made of different gratings (various opto-geometrical parameters). This device is then characterized using quadri-wave lateral shearing interferometry in the LWIR domain (integration over the 8–12 µm range). This technique is the first one, to our best knowledge, which allows a spatially-resolved evaluation of both amplitude and phase in a single measurement. This is validated by a preliminary experiment, using a broad-band source and under normal incidence. Comparisons have been made with amplitude measurements using a FTIR spectrometer, and numerical simulations of the induced phase shift: they show a good agreement.

This technique opens up the way for a complete (spectral and directional) characterization bench for sub-wavelength gratings. The spectral resolution requires the use of broad-band tunable and highly luminescent sources, which may be obtained in the infrared using *e.g.* parametric conversion in non-linear optical materials [25, 26]. One may also use super-continuum generators in non-linear crystals [27] or in photonic fibers [28], provided that the spectral selection is made with an interference filter. The angular resolution is straightforward, using classical goniometric techniques [29].

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