1. Introduction

The metrology of segmented wave-fronts is a critical issue in many industrial or research domains. For instance, this type of measurement can concern the verification of discontinuous structures, e.g. diffractive optical elements, to check their conformity to given specifications; it is also necessary when the relative positioning of optical elements is critical, particularly during the phasing of segmented mirrors, like the primary mirror of the Keck telescope [1]. In the particular case of phasing segmented mirrors, many types of techniques have been developed, among those we find dedicated techniques [1, 2], modified curvature sensing technique [3], dispersed fringe sensing [4], diffraction pattern analysis [5] and also shearing interferometry using a Fourier plane filtering [6].

Lateral shearing interferometers (LSI) based on diffraction grating can also be used to evaluate segmented wave-front. For instance, the three-wave lateral shearing interferometer [7, 8] has already been used to measure a segmented wave-front generated by an etched substrate [9]. Compared to the techniques described above, lateral shearing interferometry can be used to measure the wave-front discontinuities simultaneously with the aberrations induced by the quality of individuals segments or by atmospheric turbulence. But the dynamic range is limited within ±λ/2, λ being the working wavelength.

In this paper, we propose to extend the dynamic range of the LSIs and to apply the results to the quadri-wave lateral shearing interferometer (QWLSI). The QWLSI offers the possibility to analyze the wave-front in a bidimensional way in one single measurement using polychromatic light [10, 11]. Moreover, this technique combines metrology and ease of use qualities such as compactness, simplicity or the possibility to work in presence of vibrations. It can also be integrated in an adaptive optics loop for real time correction, as it is currently done in ultra-intense laser facilities [12].

In section 2, we detail the response of LSIs to a segmented wave-front and the limitation of the dynamic range. In the sections 3 and 4, we show that the use of two different colors, monochromatic or not, allows to greatly enhance the dynamic range. For sake of simplicity, these three sections are based on LSIs which generate only two replicas of the incident wave. In the section 5, we evaluate the ability of the QWLSI to measure a Keck-like segmented wave-front in the long wavelength infrared. Finally, in section 6, we present the experimental analysis of a high-dynamic segmented wave-front in the visible domain.

2. Segmented wave-front analysis

We consider here a particular LSI based on a perfectly sinusoidal diffraction grating of period p′. This particular LSI generates only two tilted replicas and the interferogram is constituted of sinusoidal fringes which pitch p=p′/2 is independent on the wavelength. The following equation describes the interferogram when the impinging wave has a uniform irradiance I ₀ and a wave-front W(x,y) :

where k is the wave number and Δ_s,x W(x,y)=W(x+s/2,y)-W(x-s/2,y) is the optical path difference between two points of W separated by the lateral shear s in the x-direction.

The influence of the aberrated wave-front is characterized by a local phase displacement θ (x,y)=kΔ_s,xW (x,y) in the deformed interferogram compared to a perfectly sinusoidal interferogram. To recover θ (x,y), we analyze the deformed interferogram in the Fourier domain [13]. In concrete terms, we study the deformations of the harmonic peak placed at 1/p contained in the spectrum of the interferogram. For that, this peak is selected by a filtering window and shifted to the central frequency. The inverse Fourier transform of this quantity is the following harmonic signal H:

Finally, to retrieve θ (x,y) we determine the imaginary part of the complex logarithm of H. The phase displacements θ (x,y) are recovered without ambiguity only if they are originally comprised within ±π. The amplitude of the measurable path difference is then limited within ±λ/2 which also limits the dynamic of the measurable wave-fronts.

When the variations of W are slow compared to s, this ambiguity limitation can easily be overpassed. Indeed W can be approximated by a first-order Taylor serie and the phase displacement θ (x,y) becomes proportional to the shear distance s [14] and so to the distance z between the grating and the observation plane as s=2λz/p′.

However, this approximation is not suited to segmented wave-fronts. Let us, for instance, consider a wave-front W constituted of two plane segments with an h-high edge located at the position x ₀ (see Fig. 1(a)). The optical path difference Δ_s,xW is then a unit s-wide crenel, C_λ (x,y), centered at x ₀ and which height is multiplied by h as shown on Fig. 1(b).

 

Fig. 1. (a) A segmented wave-front W(x,y) constituted with two segments separated by a height h. (b) The corresponding optical path is a s-wide crenel of height h: Δ_s,xW(x,y)=hC_λ (x,y).

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Consequently, the height h can only be recovered without ambiguity if it lies within ±λ/2, whatever the shear is. This concretely limits the use of the method to segmented wave-fronts with small step heights values, and thus its utility to Keck-type applications. However, one can point out that the phase displacement θ depends on the wavelength λ : this provides a supplementary degree of freedom, which will be shown to greatly enhance the dynamic range of the LSIs.

3. Extension of the dynamic range

To increase the dynamic range of the LSI, we propose to combine two different interferograms recorded with two different wavelengths λ ₁ and λ ₂ (λ ₁<λ ₂). Using two different wavelengths is indeed a classical way to remove measurement ambiguities in interferometric techniques [15, 16, 17, 18].

In the following, we consider that the interferogram at λ_i has been obtained at a distance z_i=sp′/2λ_i in order to keep the same lateral shear s. Figure 2 shows the intensity profiles obtained at the two wavelengths. A portion of the interferogram is laterally displaced where Δ_s,xW≠0. In this portion, the phase displacement is a constant equal to θ_i=h2π/λ_i (i=1, 2). We will then use of the dependence of θ_i on the wavelength to increase the dynamic range.

 

Fig. 2. Schematic profiles of two lateral shearing interferograms obtained with two different wavelengths λ₁ and λ₂ during the analysis of the segmented wave-front shown on fig. 1(a).

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In the monochromatic case, the step height is found by studying the displacement of the deformed interferogram with regard to a reference sinusoid. In the two-wavelength case, we use one deformed interferogram as a reference, and study the relative displacement of the other to it. It is equivalent to measure the difference, noted θ _s, between the angles θ ₁ and θ₂ :

where the synthetic wavelength λ_s is equal to:

By correctly choosing λ₁ and λ₂, one can easily tune the dynamic range of the measurement. Providing that h is comprised within ±λ_s/2, it can be recovered without ambiguity thanks to the use of two wavelengths.

To get θ_s, we combine the two monochromatic harmonic signals $H_{{\lambda }_1}$ and $H_{{\lambda }_2}$ obtained by the filtering process detailed above. We compute then a synthetic harmonic signal $H_{{\lambda }_s}$:

where the asterisk designates the complex conjugate. By taking the imaginary part of log ($H_{{\lambda }_s}$), one finally gets the step height h.

The two-wavelength method allows a measurement with a high dynamic but as the sensitivity varies inversely with the dynamic, this method is not suited when high sensitivity is required, to measure small variations inside the segments or small step height. To overcome this limitation, one may combine a two-wavelength measurement to obtain the step height and a monochromatic measurement to obtain the small variations with a high sensitivity. Other strategies, using more than two wavelengths, may also be applied.

Finally, in order to obtain the original wave-front, another optical path difference Δ_s,yW along another direction y is required [19]. Once obtained, these optical path differences Δ_s,xW and Δ_s,yW can be combined with a least-square method to yield an estimation W^e of the wave-front under study [20].

4. Two-color analysis

The two required wavelengths can be obtained by filtering a polychromatic source. To evaluate the impact of the spectral width on each interferogram, we consider a gaussian filter having a spectral density $B_{{\lambda }_1}$:

where λ₁ is the mean wavelength, $B_{{\lambda }_1}$ is the maximal transmittance and σ is the standard deviation.

The polychromatic interferogram $I_{{\lambda }_1}$,σ can then be written as an incoherent sum of monochromatic interferograms:

Where all the crenels are superposed, the polychromatic harmonic is given by:

Then we change λ by λ-λ₁ and consider that σ is smaller than λ (typically σ/λ₁<10%). In this case, the polychromatic harmonic becomes:

This expression is strongly related to the monochromatic harmonic obtained at ${\lambda }_1\left(H_{{\lambda }_1}=I_0\exp [2i\pi h⁄{\lambda }_1]\right)$:

where $F_{{\lambda }_1,\sigma ,h}$ is a real factor given by:

The step height h can then be retrieved by the method proposed in section 2. The factor $F_{{\lambda }_1,\sigma ,h}$ does not impact the measured value of h but it occurs in the height of the harmonic and consequently in the signal-to-noise ratio (SNR) of the measurement. When σ increases, the harmonic height first increases due to the higher number of photons passing the filter. Then for higher value of σ, the interogram is getting blurred due to the superposition of many phasedisplaced monochromatic interferograms and consequently, $F_{{\lambda }_1,\sigma ,h}$ decreases.

The transition is done at an optimal value σ_opt for which the harmonic height is maximal:

σ_opt depends also on the step height h. The blurring of the interferogram occurs indeed faster when h is high which has for effect to decrease σ_opt. If the range of the step height h value is approximatively known, one can choose σ in order to maximize the height of the harmonic and so the signal-to-noise ratio of the measurement. As an example, we plotted the variation of $F_{{\lambda }_1,\sigma ,h}$ versus σ when h is equal to 50µm and λ₁ is of 10µm on Fig. 3. This curve is maximal at an optimal value of σ equal to 0.32m.

 

Fig. 3. Evolution of $F_{{\lambda }_1,\sigma ,h}$ versus σ when h=50µm and λ₁=10µm.

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To extend the dynamic range, another filter, centered on a slightly different wavelength, can be used to obtain the second interferogram. These two filters can also have a common zone without changing the result. For instance one can use an interferometric filter which central wavelength can be easily selected by tilting the filter.

The implementation of the two-color method, using large band light, instead of the two-wavelength method can be particularly relevant when the incident flux is weak. Among these particular applications, we particularly think of the co-phasing of a segmented mirror of a telescope, like in the Keck telescope, using star light. The following sections describes the analysis of such a segmented wave-front with a QWLSI.

5. Application to a Keck-like wave-front

The ability of the LSIs generating two tilted waves to measure large step height is transposable, almost identically, to the QWLSI which diffracts four tilted waves. The only point is that, in the case of the QWLSI, the optical path difference along the x- and y-axis are not a perfect crenel as described in section 2.

In fact, Δ_s,xW (x,y) is the average of Δ_s,xW (x,y+s/2) and Δ_s,xW (x,y-s/2), the differences between two sheared replicas along x at two vertical values y+s/2 and y-s/2, and similarly for the y-direction. But, as the following results show, the particular pattern of the optical path differences along x and y does not significantly affect the analysis of segmented wave-fronts. For more details on the QWLSI, see Ref. [9] and [10].

To demonstrate the ability of the QWLSI to measure segmented wave-front, we have simulated the analysis of a Keck-like wave-front in the long wavelength infrared domain with wavelengths at λ ₁=10µm and λ ₂=11µm (λ_s=110µm). The analyzed wave-fronts constituted with 36 segments (see fig. 4(a)), assumed perfectly plane, each having a different piston value. After reflection, the amplitude of the piston value to be analyzed is here comprised between -20.7µm and +9.3µm. The greatest piston difference, in absolute value, occurs in the x-direction and is equal to 18.9µm. The proposed QWLSI generates about 9 measurement points along the horizontal major diagonal of a single hexagonal segment and the chosen shear distance is equal to twice an interferogram period which is equivalent to about 20% of a segment. The interferogram is then sampled on 256*256 pixels and filtered in order to simulate a square pixel filtering. The obtained interferogram at λ₁ is shown on Fig. 4(b).

The analysis process of the QWLSI interferogram is similar to the one explained in section 3 for the LSIs generating only two replicas. After the spectral analysis of the interferogram, we have the cartographies of the optical path differences Δ_s,xW and Δ_s,yW at the 84 intersegment edges. The reconstructed wave-front is shown on fig. 4(c). The global pattern of the segmented wave-front is recovered although the reconstructed segments are not perfectly plane, particularly at the vicinity of the segment edges. To compare the original and the reconstructed value of pistons, we calculated the average of the reconstructed segments on a smaller hexagonal surface (80% of the original segment surface). The curves shown on fig. 4(d) represent the original values (dashed line) and the reconstructed values (solid line). The two-color method allows to extract high-value steps. However, there remains discrepancies between the original value and the reconstructed one, we can explain them by two main reasons. At first, we do not know the derivatives values at the pupil edges; in fact, we only know 84 edges values over 132, that is to say less than 65% of the useful information. This lack of information disturbs the reconstruction process which considers the wave-front globally. The second reason is given by the high dynamic of the two-color method and consequently its poor sensitivity, contrary to the monochromatic method which can measure small wave-front differences with high sensitivity in a poor dynamic range. Consequently, in order to complete the study, one may use a one-color analysis. The combination of the two-color and one-color measurments finally provides a high dynamic and simultaneously a highly accurate study. It has to be noticed that the one-color analysis allows also to recover an additional turbulent phase, as it is a classical technique for adaptive optics [12] and also the possible defects inside the segments.

The simulation detailed above was made with a theoretical infinite SNR. To check the robustness of the QWLSI to noise, we added a Gaussian noise in order to generate an arbitrary SNR equal to height. The aim of this simulation is to show that our technique is not ill-conditioned and is able to work with noisy measurements. Figure 4(e) shows the reconstructed wave-front and Fig. 4(f) shows the mean piston values inside the segments. In the particular case studied here, the standard deviation of the difference between the original and reconstructed values is equal to 2.51µm which is similar with the value calculated on the noiseless data (1.38µm). Despite a poor amount of useful information and noise, the measurement so remains exploitable.

 

Fig. 4. The two-color analysis has been applied to the wave-front presented on (a). The fig.(b) shows a simulated monochromatic interferogram generated by a QWLSI with an infinite SNR. The fig.(c) and (e) represent the reconstructed wave-fronts with respectively the noiseless interferogram and the noisy interferogram (SNR=8). The fig.(d) and (f) represent the average piston values of the 36 segments calculated on the original wave-front (dashed line) and on the reconstructed wave-front (solid line) shown on (c) and (e).

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6. Experimental measurements

This section aims at validating the use of two colors to increase the dynamic range of lateral shearing measurement thanks to experimental analysis with a QWLSI [9, 10, 11]. The experiment described here is focused on the analysis of a segmented wave-front generated by an etched substrate in the visible domain. The wave-front W generated by the etched zone, here a phase chessboard, is equal to h=1093nm.

The two monochromatic interferograms were made with the commercial Phasics-SID4 [21], a QWLSI dedicated to the visible wavelengths. The QWLSI used here is a bidimensional grating which generates interferogram constitued whith bright spots disposed on a cartesian grid and which pitch p′ is equal to 29.6µm. The substrate was illuminated successively by two HeNe lasers at λ ₁=594nm and λ ₂=633nm (λ_s=9.64µm) and is conjugated with the detector of the QWLSI thanks to an afocal system with a unit magnification, as described on fig.5.

 

Fig. 5. Schematic test bench dedicated to the two-color analysis of etched substrates by quadri-wave lateral shearing interferometry.

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The shear distance is equal to 128µm, that is to say around four times the interferogram period, and the lateral dimension of the etched surface under test is equal to 3.8mm. Figure 6(a) shows the experimental interferogram, at λ ₁, obtained in these conditions. Combined with the other interferogram at λ ₂, two different types of information can be recovered : the steps height, at the segments edges, and the inner defects of the segments elsewhere.

Figure 6(b) represents the reconstructed wave-front from the two monochromatic interferograms, the global pattern of the phase chessboard is recovered. Figure 7 is the profile of this wave-front obtained along the line indicated on Fig. 6(b) and shows that the step, equal to 1093nm is recovered inside the segments and also that there are artifacts at the segment edges. As these artifacts has a specific high-spatial frequency pattern, they can be removed with image processing techniques, like a low-pass filter for instance.

This experiment was a first step to demonstrate the ability of a QWLSI to measure segmented wave-fronts having plane segments. Further work will particularly aim at avoiding the artifacts on the segment edges. For that, we are especially working on the adaptation of the Phasics-SID4 in order to have a dedicated set-up for two-color analysis.

7. Conclusion

The ability to measure segmented wave-fronts thanks to lateral shearing interferometry techniques was detailed. The monochromatic analysis was shown to limit the dynamic range of measurable heights between two segments. In fact, this method can only measure the step height whithin ±λ/2, λ being the working wavelength, without ambiguity. To overpass this limitation, we shown that the combination of two monochromatic analysis can greatly extend this poor dynamic range. The proposed method can also be applied with polychromatic light with significantly large bandwidth.

The applications concerned by this method are varied, among them we can cite the analysis of diffractive elements, the phasing segmented mirrors or also the phasing of bundle of single-mode fibers (for coherent beam combining). For these applications, we especially think of the use of the quadri-wave lateral shearing interferometer. The QWLSI is indeed particularly interesting to cartography a wave-front because it gives bidimensional information on it, the optical phase differences along x and y, in one single measurement. To demonstrate the ability of a QWLSI to measure bidimensional segmented wave-fronts, we first computed its response during the analysis of a Keck-like wave-front, with ideal and noisy interferograms. In this case, thanks to the two-color method, the original wave-front was recovered without ambiguity and with an error allowing a mono-color analysis. A second step was dedicated to the experimental analysis of a segmented wave-front generated by an etched substrate. This study confirms the ability of the QWLSI to measure high dynamic steps but also point out the apparition of artifacts at the edges of the reconstructed wave-front. Further work will be focused on the resolution of this effect by designing a new interferometer and developping dedicated image processing techniques.

 

Fig. 6. (a)The experimental interferogram at λ₁. The axis x and y are given by the orientation of the bright spots. (b)The reconstructed wave-front thanks to a two-color analysis.

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Fig. 7. A profile of the reconstructed wave-front along the line drawn on fig. 6(b).

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