Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles

We propose an experimental set-up working in the optical domain for determining the phase of the field diffracted by an object. We show that our apparatus can be used to reconstruct the deterministic profile of rough surfaces. We consider the particular case of diffraction gratings.


Introduction
Diraction is known to be a powerful method for the non-destructive and remote characterisation of targets. Indeed, measuring both amplitude and phase of the ®eld diracted by an object, gives precious information on its optogeometrical parameters such as its permittivity distribution or its shape. This technique is used, for example, in the radiofrequency domain to solve deterministic problems like the reconstruction of surface pro®les or embedded objects [1]. In principle this method can be applied to any range of frequencies, depending on the size of the target. However, as an interferometric apparatus is required, determining the phase of the ®eld in the optical domain is a dicult task. For this reason, most eorts have been devoted to the study of the diracted intensity pattern. Although this information gives access to the second order (spectral density) statistical properties of the diracting structure [2], the inverse deterministic problem when not using an optimisation method [3,4] requires phase measurements of the diracted ®eld.
The problem of detecting the phase of the ®eld diracted by an object has been addressed for different kind of applications. For example for chemical sensing applications that require the detection of small variations of permittivity, interferometric con®gurations have been proposed in order to measure the phase of the re¯ected beam near the electromagnetic resonances [5,6]. To separate surface contribution and volume contribution to electromagnetic scattering, angle-resolved ellipsometric scatterometers have been developed [7,8]. In this case the polarimetric phase of the scattered ®eld is measured. An experimental con®guration [9] has recently been proposed to directly measure both amplitude and phase in a scanning near-®eld optical microscope [10]. The underlying principle is interference between a probe beam and a reference beam. However, extending this technique to far-®eld measurements is not straightforward. Indeed, in this case, accurate thermal and mechanical stabilisation is needed.
In this paper we present an experimental set-up which gives access, within the validity of Kirchho approximation, to the phase of the diracted ®eld in the incidence plane. The crucial advantage of our solution is its insensitivity to mechanical drifts or thermal¯uctuations. We show that our technique can be used for reconstructing pro®les of rough surfaces with a straight inversion of the experimental data i.e. without using any optimisation method. We illustrate this result by considering the particular case of periodic surfaces.

Principle of measurement
The principle of phase measurement of the ®eld diracted from a rough interface between two homogeneous media lies in the coherent mixing of two beams coming from the same laser source which overlap on the sample surface. The incidence angles of the two beams can be chosen independently as well as the observation direction of the scattered ®eld. In a given direction, this scattered ®eld results from the interference between the speckle patterns produced by the two incident beams. The phase information is measured by means of a synthetic heterodyne detection scheme. A photodetector rotating around the sample permits one to record the angular variations of the phase in the incidence plane. The phase is measured with respect to a reference signal given by a ®xed detector. Light from a helium±neon laser (k 633 nm) is divided into two beams with the use of a beam splitter. The beams form the two arms of the interferometer (see Fig. 1). Light in one arm is coupled in a polarisation-maintaining optical ®bre. The end of the ®bre can rotate around the sample and the collimated output beam illuminates the sample. The other beam, which lies in the same plane as the former, passes through an electro-optic phase modulator (EOPM). The two beams of the interferometer are superimposed on the surface of the sample. Introducing an optical path delay allows one to set the optical path difference between the two arms of the interferometer to a value which is smaller than the coherence length of the source. The direction of polarisation of the incident light can be switched from TE to TM by rotating simultaneously the polariser, the EOPM and the optical axis of the ®bre. An optical density is used to vary the intensity balance between the two beams overlapping on the sample surface. Rotating the sample around a vertical axis can vary the incidence angles of the two incident beams.
A saw-tooth voltage modulation whose amplitude corresponds to a 2p phase shift is applied to the EOPM. Thus the interference signals obtained in all directions of space have a sine shape in time. In direction h the scattered intensity can be written as: where i 1 and i 2 are the incidence angles of beams 1 and 2, respectively. uh; i 1 and uh; i 2 are the phase shifts of the respective beams resulting from scattering from the surface sample in direction h. / 0 t is the saw-tooth signal generated by the EOPM. The optical density is chosen in order to maximise the modulation amplitude of the interference signal recorded on the reference detector. Phase dierence Dn corresponds to the phase delay between the two incident beams, accumulated from the beam splitter to the sample surface. Obviously, any mechanical drift and any thermal uctuation in the set-up, which aects the optical path delays of the two incident beams, causes Dn to vary, making the measurement of Ih highly unstable. In order to avoid any complex stabilisation procedure, we propose a con®guration with two measurement channels. The two outputs are processed to form a signal which is independent of Dn.
The two photodetectors placed in scattering directions h and h H , are connected to a lock-in ampli®er working in the phase detection mode. The measured phase / is given by: which is independent of Dn and thereby insensitive to instabilities in the set-up. Away from plasmon anomalies [11], when working at nearly normal incidence, for small emergence angles and within the domain of application of Kirchho approximation (for theoretical developments see Section 3) / can be written as follows (u can be written a function of r, r H , r 1 and r 2 ): where r 2p=k sin h, r H 2p=k sin h, r 1 2p= k sin i 1 and r 2 2p=k sin i 2 .
For h H ®xed and varying angle h, if i 1 and i 2 are such as r 2 À r 1 Dr the term / n given by: where C is such as r H À r 1 C Dr, r 2 À r 1 Dr and r À r 1 n Dr, can be calculated for all n P ÀN =2; N =2. The discrete integration of phase terms / n permits one to determine the phase shift u n un Dr corresponding to a scattering angle h n arcsink=2pn Dr, to within a linear drift. However, as we will see, this drift has no in¯uence on the reconstruction. Thus we can obtain the phase of the diracted ®eld at angles h n , the angular sampling rate being arbitrarily small. When the sample is illuminated by only one incident beam, the set-up is equivalent to a classical angle-resolved scatterometer. For a ®xed angle of incidence i 1 , the intensity can be measured as a function of h (of r) and the complex amplitudes of the optical waves scattered in the incidence plane can be determined.

Pro®le reconstruction
Our set-up gives access to the value of the complex amplitude of the diracted ®eld in the incidence plane. We have applied our method of measurement to the problem of pro®le reconstruction. For the sake of simplicity, experiments have been performed with two metallic gratings. The samples are aluminium ruled gratings with 150 grooves/mm by Jobin-Yvon and with a grating height h < 200 nm. With these characteristics the grooves are shallow enough to ensure the validity of Kirchho approximation. Notice that, in normal incidence, 21 orders are diracted from the gratings at wavelength k 0:633 lm. Detectors are silicon photodiodes working in photovoltaic regime placed in order to intercept the diracted orders of interest. Measurements have been made by varying h (and then r) in all diracted orders with Dr 2p=d (see Eq. (4)) where d is the grating period. Incidence angles i 1 and i 2 angles are chosen in order to avoid plasmon resonances that can occur with metallic surfaces illuminated in TM polarisation. This can be checked by measuring beforehand the re¯ectivity of the grating as a function of the angle of incidence.
For the reconstruction of the pro®le from the scattered ®eld, the physical optics (Kirchho) approximation has been used. The latter is known to be valid at non-grazing incidences for surfaces that are``smooth'' on the scale of the wavelength. This goes for surfaces whose radius of curvature is much greater than the wavelength (see e.g. Ref. [12] for a thorough discussion). For 1D perfectly conducting surfaces, if h designates the scattering angle and i designates the incidence angle, the scattered amplitude in the Kirchho approximation can be written as: where b 2p=k cos h, b i 2p=k cos i and A 0 is an optical factor depending on the polarisation. Note that in the case of an echelette grating, the Kirchho approximation is not valid near the edges. However, since the grating period is much greater than the wavelength, we will neglect the edge diraction. We will also assume the grooves to be too shallow to cause multiscattering phenomena. These two assumptions ensure the validity of the Kirchho approximation. Working at normal incidence and small emergence angles, one has cos i % cos h % 1, so that the Kirchho approximation is reduced to the Fourier transform of the exponentiated pro®le (Fraunhofer approximation): Thus, the pro®le inversion is obtained from a bare inverse Fourier transform of the diracted far ®eld. Note that Eq. (7) shows that the modulus of the inverse Fourier transform of Sh; i is equal to 1. This can be used to verify the accuracy of the experimental results. In principle there exist a more accurate procedure for pro®le retrieval for the Kirchho approximation [13], but we found it hardly tractable at the experimental level since it requires simultaneous variations of the incidence and emergence angles.

Validity of the approximation used
In order to check numerically the validity of the approximation used, we have compared different initial pro®les to those reconstructed from Eq. (7). For the comparison, the complex amplitudes Sh; i have been calculated with a rigorous method based on the dierential formalism [14,15]. The comparison has been drawn for an echelette pro®le and for a sinusoidal one. For the echelette pro®le the blaze angle a was chosen as a 3°and the depth as h 348 nm. The depth of the sinusoidal pro®le is h 300 nm. Fig. 2(a) shows the results obtained for these two particular cases. One can see that with this value of h both pro®les can be determined accurately. Nevertheless, as expected (see Section 3), a slight disagreement is obtained near the edges of the echelette pro®le. One can notice (Fig. 2(b)) that a disagreement between the reconstructed pro®le and the initial one leads the modulus of the inverse Fourier transform to be dierent from unity. Thus, studying this modulus gives precious information about the accuracy of the result. Fig. 3 gives the mean distance 1=N P jh ini x i À h rec x i j between the initial pro®le (echelette or sinusoidal) and the one determined from Eq. (7), as a function of the grating's depth. One can see that an error less than 3 nm is obtained for values of h < 300 nm.

In¯uence of the errors of measurement
We have also checked the stability of the reconstruction method of the inversion in relation to the experimental noise. We have successively introduced, on the numerical data, a 5°phase random noise and a 5% random noise on the modulus of Sh; i. Fig. 4 shows the root mean square difference between the initial pro®le (echelette or sinusoidal) and the one determined from Eq. (7). One can see that such noises lead to a mean error smaller than 2 nm on the surface pro®le. This result shows the robustness of the inversion method used.

Experimental results
The arrangement of the optical experiment is shown in Fig. 5. Fig. 6 gives the comparison of the pro®les as measured by atomic force microscopy (AFM) and the pro®les obtained after reconstruction. AFM measurements of the surfaces have been performed with a contact tip on a 50 lm 2 area, with a sampling interval of 166 nm. The AFM curves drawn in Fig. 6 are the mean pro®les along the invariance direction. In the optical experiment the beam size was 6 mm 2 . For both gratings the Fig. 3. Numerical simulation: root mean square distance between the grating pro®le and the reconstructed pro®le as a function of grating's depth h. Note that for the echelette grating the apex angle remains equal to 90°and the blaze angle increases with the grating's depth.  optical experiment was performed by rotating one detector in order to successively intercept all the diracted orders. This means that n P À10; 10 (see Eq. (4)). The grating's height is about 200 nm for the echelette grating and the defects of ruling are clearly visible. The mean distance between the result obtained by AFM and the reconstructed pro-®le is less than 7 nm. The height of the sinusoidal pro®le is about 60 nm and the mean distance between curves is 2.5 nm. In the reconstruction procedure the gratings' pro®les are assumed to be invariant within the beam area. The discrepancy between the AFM data and the reconstructed pro®les can be attributed to the partial validity of this assumption. Note that the same results are obtained for TE and TM polarisations. This con-®rms the validity of the scalar approximation used.
To show the necessity of measuring the phase in reconstruction problems, the inversion has been performed by replacing the measured phases with a random phase (dotted lines in Fig. 6(a) and (b)). The pro®les obtained this way do not have the same shape as the AFM-measured pro®les. The necessity of measuring the phase is more clear in the case of a sinusoidal pro®le. For the echelette pro®le the reconstructed pro®le resembles however to an echelette pro®le, which is a generic phenomenon when eciency in one order is very strong. Indeed, this implies a quasi-linear phase for the exponentiated pro®le. Fig. 7 gives the modulus of the Fourier transform (see Eq. (7)) of the experimental values of Sh; i. Following the remark of Section 4 these results con®rm the accuracy of the experimental data.

Conclusion
We have shown the possibility of measuring the phase of the diracted ®eld with an interferometric set-up insensitive to instabilities. The angular phase information, which is obtained from a small number of measurements, can be used for reconstructing grating pro®les. The inversion method described can be generalised to the case of rough surfaces whose optogeometrical parameters are so that Kirchho approximation can be used. Dielectric surfaces may also be studied with a suitable theory.