- Email: [email protected]
Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes
Optics Communications 279 (2007) 240–243 www.elsevier.com/locate/optcom
Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes Bruno Toulon
a,*
, Je´roˆme Primot a, Nicolas Gue´rineau a, Riad Haı¨dar a, Sabrina Velghe b, Raymond Mercier c
a
c
Office National d’Etudes et de Recherches Ae´rospatiales, Chemin de la Hunie`re, F-91761 Palaiseau Cedex, France b Phasics SA, XTEC, Campus de l’Ecole Polytechnique, F-91128 Palaiseau, France Laboratoire Charles Fabry de l’Institut d’Optique, Campus Polytechnique, RD 128, F-91127 Palaiseau Cedex, France Received 10 April 2007; received in revised form 29 June 2007; accepted 30 July 2007
Abstract We present an original step-selective mode which allows to measure only the steps and not the slowly varying aberrations of a wave front. This mode can be implemented when measuring segmented wave front by a diffraction-grating-based lateral shearing interferometer. This set-up rests on the different chromatic response of these interferometers depending on the rate of change of the impinging wave front: for smooth defects, the response is classically achromatic whereas it is chromatic for a step variation, which was to our knowledge overlooked. The interest of this mode for astronomical measurements is highlighted. First we present theoretical considerations to show how this mode of measure is possible; then a numerical simulation illustrates them. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Hz; 42.62.Eh; 42.79.Dj; 42.68.Wt Keywords: Wave front measurement; Lateral shearing interferometry; Alignment; Astronomy
Lateral shearing interferometers (LSIs) are commonly used to analyze wave fronts in order to test optical components or laser beams. Since the wave front under study is coherently superposed to a laterally shifted replica of itself, this technique does not require a reference wave, which is a major advantage in terms of measurement simplicity [1]. In this article, we dwell in particular upon the measurement of segmented wave fronts. Since Ronchi’s historical article [2] diffraction-grating-based LSIs are believed to be perfectly achromatic; however, as we shall show below, an interesting chromatic mode can be pointed out when measuring steps. A segmented wave front is made of the discontinuous apposition of continuous segments (see Fig. 1). In other words, it means a continuous surface which presents dis-
continuities. Such a wave front can be generated by the apposition of beams created by several optical elements: the segmented mirror of a telescope for example [3] or fibers for laser beam coherent recombination [4]. Many techniques have been developed to measure the height of steps [5,6] and most were specifically designed to phase segmented telescope mirrors: modified Shack-Hartmann [3], Foucault test [7], dispersed fringe sensing [8], curvature sensor [9], etc. Generally speaking, diffraction-based LSIs replicate the impinging wave front in two or more replicas (see Fig. 2), which are shifted by the lateral shearing distance s. Diffraction-based LSI are then characterized by two properties: (A) since s is directly linked to the diffraction by a grating, it is proportional to the wavelength, according to the formula hereafter:
*
Corresponding author. Tel.: +33 169936338; fax: +33 169936345. E-mail addresses: [email protected] (B. Toulon), [email protected] (J. Primot). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.07.037
s¼
2kz ; p
ð1Þ
B. Toulon et al. / Optics Communications 279 (2007) 240–243
241 Wstep
h
continuous
0
LSI
continuous
kh
ϕ
Fig. 1. Example of a segmented surface.
difference
s
wavefront W
s
Fig. 3. Wave front Wstep generates the phase u, represented with its replica in dashed line. The phase difference between the replicas is then computed and plotted.
diffraction grating
u first replica
second replica
s Fig. 2. An impinging wave front W is divided in two replicas separated by a distance s along the u-axis.
where k is the working wavelength, z is the observation distance and p the grating period; (B) after computation, LSIs leads to the difference between the phases induced by the replicas; these phases being proportional to the wave number k and thus inversely proportional to the working wavelength. Now that we have these two generic properties, let us consider a continuous wave front at the wavelength k. We suppose that the wave front is slowly varying compared to the average lateral shearing distance s and we call it Wslow. The difference between the replicas (property B) tends classically to the derivative of the wave front (except from the possible non-derivable points), which is the firstorder approximation: s s oW slow kW slow u þ þ oðuÞ; ð2Þ kW slow u ¼ ks 2 2 ou where u is the direction of the lateral shift, and o(u) the neglected orders. According to the preceding paragraph, the ks product is constant whatever the wavelength (property A). Consequently, the measure of a slowly varying wave front, such as Wslow, is achromatic. On the contrary, let us consider a pure step wave front Wstep represented by the Heaviside step function of height h (see Fig. 3). In this case, the difference between the replicas is a s-wide crenel whose height is kh; it thus leads to a chromatic measure. The first-order approximation detailed above cannot be made anymore, however in a sense the difference approximates the derivative, which is a Dirac function, by a crenel. A reconstruction of the impinging wave front by integration, thanks to iterative Fourier transforms for instance [10–12], is still possible. The integration of the derivative is then approximated
by the integration of the difference between the replicas. This is geometrically equivalent to the computation of the surface of the phase difference, which is given by the product ks. This surface is then achromatic as the product is achromatic. Moreover it implies that the same reconstruction algorithm can be used for the slowly varying defects and for the steps. The chromaticity of LSIs can be profitably used to overpass the limitations of dynamic when measuring steps [13]. In optics the classical method is to use several wavelengths [1,14]: two interferograms can be grasped around two wavelengths ki, so as to measure optical path differences larger than ±ki/2. As the measure of steps is chromatic, this method can be applied, whereas it is impossible on the slowly varying parts of wave front. The combination of the two preceding defects (for instance Wstep + Wslow as pictured in Fig. 1) leads to a segmented surface, as defined above. In this case, both chromatic and achromatic regimes coexist. This allows another major application of the chromatic mode is the selection of the fast varying defaults and the cancelation of the slowly varying ones. Let us call Di the difference between the phase replicas at the wavelength ki. Then when one computes the difference Di Dj at two wavelengths ki and kj, only the chromatic parts, that is the rapidly varying ones, remain. This can be written h s s i Di ¼ k i W slow u þ W slow u h 2 s 2 s i þ k i W step u þ W step u 2 2 h oW slow s s i ¼ k i si þ oðuÞ þ k i W step u þ W step u ; 2 2 ou ð3Þ thus
h s s i Di Dj ¼ ðk i k j Þ W step u þ W step u : 2 2
ð4Þ
For instance, if the measure is done on a segmented wave front presenting aberrations on the segments, only the steps between the segments are computed; the aberrations of the inner segments will not be taken into account. Consequently the two working modes described above can be summed up: on the one hand a LSI measures the
242
B. Toulon et al. / Optics Communications 279 (2007) 240–243
derivative of a wave front when the characteristic distance is long compared to s, and on the other hand it measures the defect itself, that is the step height. In concrete terms, we illustrate these considerations by a numerical simulation (see Fig. 4). A surface is made from a randomly generated continuous surface, which illustrates turbulence-induced aberrations, of amplitude 2k1 with k1 = 11 lm, stepped diagonally (Fig. 4a). The step height is k1. We then simulate an interferogram (Fig. 4b) produced by a LSI based on the interference of two tilted replicas [2,15]. The same reasoning could be done with two parallel beams [16,17]. The interferogram is simulated at two wavelengths (k1 = 11 lm and k2 = 10.5 lm) and is used to compute the difference quotients of the wave front at each wavelength (Fig. 4d for one wavelength), thanks to a demodulation technique [12,18] (Fig. 4c). On the bottom of the figure (Fig. 4e and f), the difference of the two difference quotients obtained at the two wavelengths are pictured: only the discontinuous parts remain and the continuous parts, which are nearly identical, are canceled. As a preliminary experiment, we present here the results obtained for a slightly different configuration: the interferograms were obtained in the particular case of a visible quadri-wave lateral shearing interferometer (QWLSI). The grating, instead of producing two replicas of the
impinging wave front, produces four tilted replicas in a Cartesian geometry. This device is amply described in [19–21,13]. This implementation of LSI offers the possibil-
a
b
c
Fig. 4. (a) Impinging wave front; (b) zoom of a part of the simulated interferogram showing the discontinuity and (c) its spectrum (logarithmic); (d) derivative of the wave front at k1 deduced from the interferogram; (e) difference of the two derivatives computed at the two wavelengths k1 and k2; (f) profile along the dashed line of the difference of derivatives.
wave front
grating Detector
Fig. 5. (a) Experimental interferogram at k1; (b) difference quotient from the measured wave front at k1; (c) profile of the difference quotient computed at the two wavelengths k1 and k2 and of the difference of them. Note that the difference has been vertically shifted for clarity.
Fig. 6. Schematic experimental bench used in the experimental demonstration of the step-selective mode.
B. Toulon et al. / Optics Communications 279 (2007) 240–243
Fig. 7. Integration of the difference quotients at ki (i 2 {1, 2}) and integration from the difference (step-selective mode).
ity to analyze the wave front in a bidimensional way, and thus compute the derivatives or difference quotient in two directions in the same measure. Consequently, the interferogram is made of bright points instead of fringes. So as to be closer of the numerical simulation we only extract one derivative of the interferograms. We used two He–Ne lasers at k1 = 594.1 nm and k2 = 632.8 nm. The analyzed surface is a phase chessboard which height is 425 nm. We used a commercial QWLSI [22]. The experimental set-up is represented in Fig. 6: the wave front under study is optically conjugated with the detector of the QWLSI thanks to an afocal system. The results are presented in Fig. 5. The interferograms at the two wavelengths (Fig. 5a for the first wavelength) are used to compute the difference quotients (Fig. 5b), and finally the difference of the two difference quotients is computed; it shows that the slowly varying aberrations are highly reduced: the tilt of the derivative in the continuous parts is canceled when computing the difference, and so is the residual, slowly varying aberrations. The difference of the difference quotients is indeed mostly equal to zero and horizontal. Note that the processing of these experimental results are still under construction; for the moment we are not able to retrieve the exact height of the difference quotients. To illustrate the step-selective property of this experiment, we have integrated the computed difference quotient (Fig. 7). This integration leads to the profile of the measured surface. The integrated profile obtained with the difference of the difference quotients (step-selective mode) shows that the aberrations in the continuous parts are not taken into account in the reconstruction. Only the step is visible. The step-selective measurement mode can be of great help in astronomy, where segmented mirrors of large telescopes required to be aligned to minimize the wave front errors induced by the mirror itself. The classical methods
243
to evaluate steps measure both steps and turbulenceinduced aberrations whereas the proposed method is, to our knowledge, the first to select and measure only the steps. The integration of the LSI is classically done by conjugating the entry pupil of the telescope with the detector plane. In conclusion, we have pinpointed the chromatic working mode of grating-based LSIs. This behaviour is all the more surprising since it can coexist with the classical achromatic mode in discontinuous surfaces like segmented wave fronts. In this sense, chromaticity of the device is not intrinsic but set by the analyzed wave front and the lateral shearing distance. These two regimes allow an original way of measuring segmented surfaces by selecting only the stepped (or fast varying) parts. This regime can be profitably used in astronomy for example, to measure perfectly stepped mirrors and leave aside turbulenceinduced aberrations. So as to experimentally illustrate and test these properties, the authors are currently developing a standard segmented surface, which should allow precise characterization of the steps and reliable intercomparison of different measurement techniques. References [1] D. Malacara (Ed.), Optical Shop Testing, second ed., Wiley Interscience, 1992. [2] V. Ronchi, Appl. Opt. 3 (1964) 437. [3] G. Chanan, C. Ohara, M. Troy, Appl. Opt. 37 (1998) 140. [4] S. Demoustier, A. Brignon, E. Lallier, J.-P. Huignard, J. Primot, Coherent combining of 1.5 lm Er–Yb doped single mode fiber amplifiers, in: Conference on Laser and Electro-Optics, CThAA5, 2006. [5] L. Deck, P. de Groot, Appl. Opt. 33 (1994) 7334. [6] S. Mirza, C. Shakher, Opt. Eng. 44 (2005) 013601. [7] S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, P. Stefanini, Opt. Lett. 30 (2005) 2572. [8] F. Shi, G. Chanan, C. Ohara, M. Troy, D.C. Redding, Appl. Opt. 43 (2004) 4474. [9] G. Chanan, M. Troy, E. Sirko, Appl. Opt. 38 (1999) 704. [10] K.R. Freischlad, C.L. Koliopoulos, J. Opt. Soc. Am. A 3 (1986) 1852. [11] C. Roddier, F. Roddier, Appl. Opt. 30 (1991) 1325. [12] J. Primot, Appl. Opt. 32 (1993) 6242. [13] S. Velghe, N. Gue´rineau, R. Haı¨dar, B. Toulon, S. Demoustier, J. Primot, Opt. Express 14 (2006) 9699. [14] C. Polhemus, Appl. Opt. 12 (1973) 2071. [15] M.P. Rimmer, J.C. Wyant, Appl. Opt. 14 (1975) 142. [16] J.C. Wyant, Appl. Opt. 12 (1973) 2057. [17] C.L. Koliopoulos, Appl. Opt. 19 (1980) 1523. [18] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156. [19] J. Primot, L. Sogno, J. Opt. Soc. Am. A 12 (1995) 2679. [20] J. Primot, N. Gue´rineau, Appl. Opt. 39 (2000) 5715. [21] J.-C. Chanteloup, Appl. Opt. 44 (2005) 1559. [22] http://www.phasics.com.
Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes Bruno Toulon
a,*
, Je´roˆme Primot a, Nicolas Gue´rineau a, Riad Haı¨dar a, Sabrina Velghe b, Raymond Mercier c
a
c
Office National d’Etudes et de Recherches Ae´rospatiales, Chemin de la Hunie`re, F-91761 Palaiseau Cedex, France b Phasics SA, XTEC, Campus de l’Ecole Polytechnique, F-91128 Palaiseau, France Laboratoire Charles Fabry de l’Institut d’Optique, Campus Polytechnique, RD 128, F-91127 Palaiseau Cedex, France Received 10 April 2007; received in revised form 29 June 2007; accepted 30 July 2007
Abstract We present an original step-selective mode which allows to measure only the steps and not the slowly varying aberrations of a wave front. This mode can be implemented when measuring segmented wave front by a diffraction-grating-based lateral shearing interferometer. This set-up rests on the different chromatic response of these interferometers depending on the rate of change of the impinging wave front: for smooth defects, the response is classically achromatic whereas it is chromatic for a step variation, which was to our knowledge overlooked. The interest of this mode for astronomical measurements is highlighted. First we present theoretical considerations to show how this mode of measure is possible; then a numerical simulation illustrates them. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Hz; 42.62.Eh; 42.79.Dj; 42.68.Wt Keywords: Wave front measurement; Lateral shearing interferometry; Alignment; Astronomy
Lateral shearing interferometers (LSIs) are commonly used to analyze wave fronts in order to test optical components or laser beams. Since the wave front under study is coherently superposed to a laterally shifted replica of itself, this technique does not require a reference wave, which is a major advantage in terms of measurement simplicity [1]. In this article, we dwell in particular upon the measurement of segmented wave fronts. Since Ronchi’s historical article [2] diffraction-grating-based LSIs are believed to be perfectly achromatic; however, as we shall show below, an interesting chromatic mode can be pointed out when measuring steps. A segmented wave front is made of the discontinuous apposition of continuous segments (see Fig. 1). In other words, it means a continuous surface which presents dis-
continuities. Such a wave front can be generated by the apposition of beams created by several optical elements: the segmented mirror of a telescope for example [3] or fibers for laser beam coherent recombination [4]. Many techniques have been developed to measure the height of steps [5,6] and most were specifically designed to phase segmented telescope mirrors: modified Shack-Hartmann [3], Foucault test [7], dispersed fringe sensing [8], curvature sensor [9], etc. Generally speaking, diffraction-based LSIs replicate the impinging wave front in two or more replicas (see Fig. 2), which are shifted by the lateral shearing distance s. Diffraction-based LSI are then characterized by two properties: (A) since s is directly linked to the diffraction by a grating, it is proportional to the wavelength, according to the formula hereafter:
*
Corresponding author. Tel.: +33 169936338; fax: +33 169936345. E-mail addresses: [email protected] (B. Toulon), [email protected] (J. Primot). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.07.037
s¼
2kz ; p
ð1Þ
B. Toulon et al. / Optics Communications 279 (2007) 240–243
241 Wstep
h
continuous
0
LSI
continuous
kh
ϕ
Fig. 1. Example of a segmented surface.
difference
s
wavefront W
s
Fig. 3. Wave front Wstep generates the phase u, represented with its replica in dashed line. The phase difference between the replicas is then computed and plotted.
diffraction grating
u first replica
second replica
s Fig. 2. An impinging wave front W is divided in two replicas separated by a distance s along the u-axis.
where k is the working wavelength, z is the observation distance and p the grating period; (B) after computation, LSIs leads to the difference between the phases induced by the replicas; these phases being proportional to the wave number k and thus inversely proportional to the working wavelength. Now that we have these two generic properties, let us consider a continuous wave front at the wavelength k. We suppose that the wave front is slowly varying compared to the average lateral shearing distance s and we call it Wslow. The difference between the replicas (property B) tends classically to the derivative of the wave front (except from the possible non-derivable points), which is the firstorder approximation: s s oW slow kW slow u þ þ oðuÞ; ð2Þ kW slow u ¼ ks 2 2 ou where u is the direction of the lateral shift, and o(u) the neglected orders. According to the preceding paragraph, the ks product is constant whatever the wavelength (property A). Consequently, the measure of a slowly varying wave front, such as Wslow, is achromatic. On the contrary, let us consider a pure step wave front Wstep represented by the Heaviside step function of height h (see Fig. 3). In this case, the difference between the replicas is a s-wide crenel whose height is kh; it thus leads to a chromatic measure. The first-order approximation detailed above cannot be made anymore, however in a sense the difference approximates the derivative, which is a Dirac function, by a crenel. A reconstruction of the impinging wave front by integration, thanks to iterative Fourier transforms for instance [10–12], is still possible. The integration of the derivative is then approximated
by the integration of the difference between the replicas. This is geometrically equivalent to the computation of the surface of the phase difference, which is given by the product ks. This surface is then achromatic as the product is achromatic. Moreover it implies that the same reconstruction algorithm can be used for the slowly varying defects and for the steps. The chromaticity of LSIs can be profitably used to overpass the limitations of dynamic when measuring steps [13]. In optics the classical method is to use several wavelengths [1,14]: two interferograms can be grasped around two wavelengths ki, so as to measure optical path differences larger than ±ki/2. As the measure of steps is chromatic, this method can be applied, whereas it is impossible on the slowly varying parts of wave front. The combination of the two preceding defects (for instance Wstep + Wslow as pictured in Fig. 1) leads to a segmented surface, as defined above. In this case, both chromatic and achromatic regimes coexist. This allows another major application of the chromatic mode is the selection of the fast varying defaults and the cancelation of the slowly varying ones. Let us call Di the difference between the phase replicas at the wavelength ki. Then when one computes the difference Di Dj at two wavelengths ki and kj, only the chromatic parts, that is the rapidly varying ones, remain. This can be written h s s i Di ¼ k i W slow u þ W slow u h 2 s 2 s i þ k i W step u þ W step u 2 2 h oW slow s s i ¼ k i si þ oðuÞ þ k i W step u þ W step u ; 2 2 ou ð3Þ thus
h s s i Di Dj ¼ ðk i k j Þ W step u þ W step u : 2 2
ð4Þ
For instance, if the measure is done on a segmented wave front presenting aberrations on the segments, only the steps between the segments are computed; the aberrations of the inner segments will not be taken into account. Consequently the two working modes described above can be summed up: on the one hand a LSI measures the
242
B. Toulon et al. / Optics Communications 279 (2007) 240–243
derivative of a wave front when the characteristic distance is long compared to s, and on the other hand it measures the defect itself, that is the step height. In concrete terms, we illustrate these considerations by a numerical simulation (see Fig. 4). A surface is made from a randomly generated continuous surface, which illustrates turbulence-induced aberrations, of amplitude 2k1 with k1 = 11 lm, stepped diagonally (Fig. 4a). The step height is k1. We then simulate an interferogram (Fig. 4b) produced by a LSI based on the interference of two tilted replicas [2,15]. The same reasoning could be done with two parallel beams [16,17]. The interferogram is simulated at two wavelengths (k1 = 11 lm and k2 = 10.5 lm) and is used to compute the difference quotients of the wave front at each wavelength (Fig. 4d for one wavelength), thanks to a demodulation technique [12,18] (Fig. 4c). On the bottom of the figure (Fig. 4e and f), the difference of the two difference quotients obtained at the two wavelengths are pictured: only the discontinuous parts remain and the continuous parts, which are nearly identical, are canceled. As a preliminary experiment, we present here the results obtained for a slightly different configuration: the interferograms were obtained in the particular case of a visible quadri-wave lateral shearing interferometer (QWLSI). The grating, instead of producing two replicas of the
impinging wave front, produces four tilted replicas in a Cartesian geometry. This device is amply described in [19–21,13]. This implementation of LSI offers the possibil-
a
b
c
Fig. 4. (a) Impinging wave front; (b) zoom of a part of the simulated interferogram showing the discontinuity and (c) its spectrum (logarithmic); (d) derivative of the wave front at k1 deduced from the interferogram; (e) difference of the two derivatives computed at the two wavelengths k1 and k2; (f) profile along the dashed line of the difference of derivatives.
wave front
grating Detector
Fig. 5. (a) Experimental interferogram at k1; (b) difference quotient from the measured wave front at k1; (c) profile of the difference quotient computed at the two wavelengths k1 and k2 and of the difference of them. Note that the difference has been vertically shifted for clarity.
Fig. 6. Schematic experimental bench used in the experimental demonstration of the step-selective mode.
B. Toulon et al. / Optics Communications 279 (2007) 240–243
Fig. 7. Integration of the difference quotients at ki (i 2 {1, 2}) and integration from the difference (step-selective mode).
ity to analyze the wave front in a bidimensional way, and thus compute the derivatives or difference quotient in two directions in the same measure. Consequently, the interferogram is made of bright points instead of fringes. So as to be closer of the numerical simulation we only extract one derivative of the interferograms. We used two He–Ne lasers at k1 = 594.1 nm and k2 = 632.8 nm. The analyzed surface is a phase chessboard which height is 425 nm. We used a commercial QWLSI [22]. The experimental set-up is represented in Fig. 6: the wave front under study is optically conjugated with the detector of the QWLSI thanks to an afocal system. The results are presented in Fig. 5. The interferograms at the two wavelengths (Fig. 5a for the first wavelength) are used to compute the difference quotients (Fig. 5b), and finally the difference of the two difference quotients is computed; it shows that the slowly varying aberrations are highly reduced: the tilt of the derivative in the continuous parts is canceled when computing the difference, and so is the residual, slowly varying aberrations. The difference of the difference quotients is indeed mostly equal to zero and horizontal. Note that the processing of these experimental results are still under construction; for the moment we are not able to retrieve the exact height of the difference quotients. To illustrate the step-selective property of this experiment, we have integrated the computed difference quotient (Fig. 7). This integration leads to the profile of the measured surface. The integrated profile obtained with the difference of the difference quotients (step-selective mode) shows that the aberrations in the continuous parts are not taken into account in the reconstruction. Only the step is visible. The step-selective measurement mode can be of great help in astronomy, where segmented mirrors of large telescopes required to be aligned to minimize the wave front errors induced by the mirror itself. The classical methods
243
to evaluate steps measure both steps and turbulenceinduced aberrations whereas the proposed method is, to our knowledge, the first to select and measure only the steps. The integration of the LSI is classically done by conjugating the entry pupil of the telescope with the detector plane. In conclusion, we have pinpointed the chromatic working mode of grating-based LSIs. This behaviour is all the more surprising since it can coexist with the classical achromatic mode in discontinuous surfaces like segmented wave fronts. In this sense, chromaticity of the device is not intrinsic but set by the analyzed wave front and the lateral shearing distance. These two regimes allow an original way of measuring segmented surfaces by selecting only the stepped (or fast varying) parts. This regime can be profitably used in astronomy for example, to measure perfectly stepped mirrors and leave aside turbulenceinduced aberrations. So as to experimentally illustrate and test these properties, the authors are currently developing a standard segmented surface, which should allow precise characterization of the steps and reliable intercomparison of different measurement techniques. References [1] D. Malacara (Ed.), Optical Shop Testing, second ed., Wiley Interscience, 1992. [2] V. Ronchi, Appl. Opt. 3 (1964) 437. [3] G. Chanan, C. Ohara, M. Troy, Appl. Opt. 37 (1998) 140. [4] S. Demoustier, A. Brignon, E. Lallier, J.-P. Huignard, J. Primot, Coherent combining of 1.5 lm Er–Yb doped single mode fiber amplifiers, in: Conference on Laser and Electro-Optics, CThAA5, 2006. [5] L. Deck, P. de Groot, Appl. Opt. 33 (1994) 7334. [6] S. Mirza, C. Shakher, Opt. Eng. 44 (2005) 013601. [7] S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, P. Stefanini, Opt. Lett. 30 (2005) 2572. [8] F. Shi, G. Chanan, C. Ohara, M. Troy, D.C. Redding, Appl. Opt. 43 (2004) 4474. [9] G. Chanan, M. Troy, E. Sirko, Appl. Opt. 38 (1999) 704. [10] K.R. Freischlad, C.L. Koliopoulos, J. Opt. Soc. Am. A 3 (1986) 1852. [11] C. Roddier, F. Roddier, Appl. Opt. 30 (1991) 1325. [12] J. Primot, Appl. Opt. 32 (1993) 6242. [13] S. Velghe, N. Gue´rineau, R. Haı¨dar, B. Toulon, S. Demoustier, J. Primot, Opt. Express 14 (2006) 9699. [14] C. Polhemus, Appl. Opt. 12 (1973) 2071. [15] M.P. Rimmer, J.C. Wyant, Appl. Opt. 14 (1975) 142. [16] J.C. Wyant, Appl. Opt. 12 (1973) 2057. [17] C.L. Koliopoulos, Appl. Opt. 19 (1980) 1523. [18] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156. [19] J. Primot, L. Sogno, J. Opt. Soc. Am. A 12 (1995) 2679. [20] J. Primot, N. Gue´rineau, Appl. Opt. 39 (2000) 5715. [21] J.-C. Chanteloup, Appl. Opt. 44 (2005) 1559. [22] http://www.phasics.com.