A compensated optical profilometer for wavefront control ... - CiteSeerX

Meas. Sci. Technol. 7 (1996) 1032–1037. Printed in the UK

A compensated optical profilometer for wavefront control of Virgo gravitational wave antenna optics Matthieu Gounelle, Vincent Loriette, Albert Claude Boccara and Roger Nahoum Laboratoire d’Optique Physique, UPR A0005 CNRS, ESPCI, 10, rue Vauquelin, 75005 Paris, France Received 20 October 1995, in final form 2 February 1996, accepted for publication 19 March 1996 Abstract. Wavefront control is crucial to the Virgo experiment. We have developed a profilometer sensitive to the sample local slope, which measures mirror profiles with nanometre sensitivity over a large frequency range, without any need for a reference surface. In order to reduce measurement errors due to atmospheric turbulence and mechanical defects, a compensation system is included as part of the bench.

1. Introduction The Virgo antenna is a complex Michelson-type interferometer working at the Nd:YAG wavelength, 1.064 µm, dedicated to the detection of gravitational waves [1]. The quality of the optical components is crucial to the success of this experiment. Virgo needs ultra-low absorption and scattering losses in the part per million range, and very well-defined mirror shapes [2]. Up to a scale of 120 mm, the mirror surface defects must not exceed 10 nm, or λ/100, peak to valley, and λ/104 RMS in the millimetre range. Those specifications are currently not achievable using standard fabrication methods. The Virgo team in charge of the realization of the optical components proposes a new coating method based on the local correction of surface defects [3]. This method requires a metrology bench usable in situ that can operate over a large spatial frequency range, with a sensitivity better than 10 nm peak ˚ for millimetre to valley for large-scale defects, up to 1 A scale roughness. Fizeau interferometers [4, 5] cannot easily attain this sensitivity over such a large frequency domain; moreover, the necessity for a reference flat imposes complex steps in order to realize ‘absolute’ measurements [6–9], Those considerations led us to develop a method based on local surface gradient measurement, the precision of which increases with spatial frequency. It has been found that a large variety of surfaces exhibit a power law topographic distribution as a consequence of their selfaffine properties [10]. This behaviour holds for optical surfaces and has been demonstrated over four orders of magnitude in spatial frequency for roughness spectra [11]. Because the Fourier spectra of our mirrors were found to follow a power law in f −α , where α ∈ [−1.6, −1.7] and f is the spatial frequency [12], this gradient measurement c 1996 IOP Publishing Ltd 0957-0233/96/071032+06$19.50

method which emphasizes the high spatial frequencies is very well adapted to Virgo’s needs. Because it uses no reference flat, it can be called an ‘absolute’ measurement. To make sure that this method is efficient enough for future needs, we have up to now used it to measure profiles. The principle of the method is described in the section 2 of this paper. In the section 3 we describe the bench that was realized. Section 4 is dedicated to the presentation of some results that were obtained on small-scale prototypes of the Virgo mirrors. 2. The principle Commercial interferometers measure the height difference z(x, y) − zref (x, y) between the sample under study and a reference surface. They cannot discriminate between the test surface shape and defects arising from the reference surface unless three flat tests are performed. We have chosen to work using a reference direction instead of a surface reference; this means that we have chosen to measure the surface gradient ∇z(x, y) instead of the altitude z(x, y). This is done by measuring the reflection angle θ (x, y) of a Nd:YAG probe laser beam incident on the surface. This reflection angle value is simply related to the local slope at the illuminated point M(x, y) and hence to the first derivative of the profile at the point M(x, y). A numerical integration enables us to reconstruct the whole profile. Because the size of the smallest defaults that we must detect is 10 nm over 120 mm, the smallest slope to which it corresponds is about 80 nr. The angular precision of our measurement must thus be better than 80 nrad. Because it is hard to attain such an angular sensitivity using a single beam, the laser beam is separated into two parts.

A compensated optical profilometer

Figure 1. The surface slope measurement bench set-up.

The first is dedicated to measurement of the sample slope and the other part defines an absolute direction. The sample global orientation cannot actually be chosen as a reference direction because during displacement it oscillates within a 10 µrad range. These oscillations are important enough to prevent measurement with the required sensitivity. We define the reference direction using a part of the laser beam and a reference mirror rigidly linked to the sample holder. 3. The method 3.1. The experimental set-up The experimental configuration for profile measurements is shown in figure 1. The sample and the reference mirror are supported by a motorized translation stage that permits us to measure a set of points {xi }1≤i≤N . The beam issued from a 15 mW TEM00 Ng:YAG laser is separated by a calcite crystal into two orthogonally polarized beams spatially separated by 2.7 mm in the vertical (x, y) plane. The beam separation is used to avoid any noise being induced by residual interferences between the two orthogonally polarized components. One of the two beams, called the ‘sample beam’, is directed towards the sample by a polarizing beam-splitter cube whereas the second beam, the ‘reference beam’, is not deflected and strikes a reference mirror after passing through an equilateral prism. The spot radius is w = 250 µm on the sample. After reflection at normal incidence from the two mirrors, the two beams return towards the cube. Two quarter-wave plates switch the polarizations so that the reference beam is now reflected by the beam-splitter and the sample beam is not. A lens of focal length f = 0.2 m focuses the two beams onto a four-quadrant detector, The lens–detector pair is an angular detector; it eliminates any spurious beam translation that could occur during the profile scanning. The two upper quadrants are used to detect the horizontal component of the reference beam deviation and the lower ones are used to detect the horizontal component of the

Figure 2. Deviation of beams by the sample slope and the orientations of the sample and of the reference mirror.

sample beam deviation. The detector is supported by a computer-controlled translation stage that permits automatic calibration. For each sample position labelled with the index i two signals are measured: the normalized ‘reference signal’ given by the reference beam Sr (i) and the normalized ‘sample signal’ Ss (i) given by the sample beam. Here normalization means that we consider the ratio of the difference and of the sum of the two diodes’ signals. Normalization of the signals eliminates any reflection factor inhomogeneities as well as laser power fluctuations. These signals are proportional to the displacements δ(i) of the respective laser spots on the detector along the x axis. The signals are low-pass filtered before being acquired by a personal computer using a 12 bit ADC card. 3.2. The principle of the measurement We have Sr (i) = Kr δr (i)

(1)

Ss (i) = Ks δs (i).

(2)

Kr and Ks are calibration constants known from a calibration procedure. Equations (1) and (2) are only valid as long as the displacements δr ,s (i) are small compared to the laser spot size on the detector. In order to determined Kr and Ks with sufficient accuracy, the calibration is performed using a motorized translation stage that supports the detector. We acquire via the acquisition board the functions Vr,s (δ) = Kr,s δ. We make a linear fit using 50 points in the zone that defines our measurement dynamics to obtain Kr and Ks . Because δ(i) = f φ(i), where φ(i) is the beam deviation angle. Then we have, as shown in figure 2, that φs (i) = 2θ (i) + 2α(i) + Ws (t)

(3)

φr (i) = −2α(i) + Wr (t)

(4) 1033

M Gounelle et al

Figure 3. A typical 4 cm mirror profile.

where α(i) is the angle between the x axis and the global orientation of the sample. The laser beam strikes the reference mirror with normal incidence and the reference mirror is adjusted so that the laser beam always strikes the same point during displacement. The reference mirror shape then makes a constant null contribution to (4). During displacement the sample’s global orientation changes and so α is a function of the point i. Ws and Wr are beam fluctuations due to atmospherical turbulence. We know that the local slope θ(i) is related to the first derivatives at point i by the relation ∂z θ (i) ∼ tan θ(i) = (5) ∂x x=xi so that

∂z φs (i) = 2 + 2α(i) = Ws (t) ∂x x=xi

(6)

φr (i) = −2α(i) = Wr (t)

(7)

and hence Sr (i) = f Kr [−2α(i)] + f Kr Wr (t) ∂z + f Ks Ws (t). Ss (i) = f Ks 2α(i) + 2 ∂x x=xi

(8) (9)

We found experimentally that α(i) and W are of the same order of magnitude as (∂z/∂x)|x=xi . It is essential to remove these contributions in order to attain the required sensitivity. Because the two beams are close to each other and follow the same path over a long distance, the two signals Ws (t) and Wr (t) are highly correlated. The 1034

amplitudes of Wr (t) and Ws (t) are each of the order 1 µrad but the amplitude difference Ws (t) − Wr (t) is less than 80 nrad. It is necessary to subtract those two signals, but as can be seen in equations (8) and (9) it is not possible to eliminate simultaneously the beam orientation fluctuations and α(i). To solve this problem we introduced an equilateral prism at a position total internal reflection. This optical component has the ability to reverse the sign of a beam deviation that passes through it. All the beam direction fluctuations that occur before the prism are reversed twice by the prism so their sign does not change. However, the deviation angle due to the reflection from the reference mirror is reversed. We have Sr (i) = f Kr [= 2α(i)] + f Kr Wr (t) ∂z + f Ks Ws (t). Ss (i) = f Ks 2α(i) + 2 ∂x x=xi

(10) (11)

We can now subtract the two signals to eliminate α(i) and Wr,s (t). So we obtain Ss (i) Sr (i) Ws (t) − Wr (t) ∂z . (12) = − − ∂x x=xi 2f Ks 2f Kr 2 The term Ws (t) − Wr (t), being much smaller than the required 80 nrad sensitivity, can be omitted in (12). So we have in the end Ss (i) Sr (i) ∂z = − = S(i). (13) ∂x x=xi 2f Ks 2f Kr S(i) is calculated numerically after the calibration has been performed.


Figure 4. The instrument sensitivity for a 4 cm profile.

3.3. The alignment procedure It is necessary to develop a definitive adjustment scheme which is independent of the sample. (i) The laser and the 45◦ mirror placed just after it are adjusted so that the beam is parallel to the translation stage displacement. This condition is essential, otherwise we would measure the reference mirror shape because during each displacement the beam would strike a different point on the reference mirror. It is this adjustment that permits definition of the reference direction. (ii) The reference mirror is adjusted so that the reference beam’s angle of incidence is 0◦ which is necessary to eliminate beam translations. (iii) The cube is adjusted so that the two spots have the same x coordinate on the detector and are vertically separated in order to work at null signal and to avoid any interferences. (iv) The prism is adjusted so that it does not break the parallelism described above. This has to be done with great care to ensure that the prism insertion does not modify Sr (i). This adjustment procedure is independent of the sample. Only the cube adjustment changes the sample beam position on the detector and this displacement can be corrected by a rotation stage that supports the detector. Thus we have defined a set-up procedure that enables us to change the sample without destroying the adjustment.

4. Profile reconstruction and signal characterization 4.1. Reconstruction of the profile The diameter under study whose profile is z(x) is samples over N points labelled with the index i, with a sampling interval 1x. We have D = N 1x. The diameter profile is obtained via a simple integration: q=j ∂z 1x X ∂z + z(j ) = z(1) + 2 q=2 ∂x x=xq ∂x x=xq−1 2≤j ≤N

(14)

where z(1) defines the altitude of the point 1. It represents the mirror spatial position which may be arbitrarily chosen equal to zero. This is equivalent to defining a mean altitude for the whole surface. It is also possible to add an arbitrary constant value to (∂z/∂x)|x=xi which is equivalent to choosing the sample tilt. (∂ z˜ /∂x)|x=xi and z˜ (i) are redefined as follows: ¯ ∂z ∂z ∂ z˜ (15) = − ∂x x=xi ∂x x=xi ∂x q=j 1x X ∂ z˜ ∂ z˜ z˜ (j ) = + 2 q=2 ∂x x=xq ∂x x=xq−1 z˜ (1) = 0

q=j ∂z 1x X ∂z + = 2 q=2 ∂x x=xq ∂x x=xq−1 q=N (j − 1) X ∂z . −1x N ∂x x=xq q=1

(16)

(17) 1035

M Gounelle et al

Figure 5. The instrument reproducibility.

In order to test the bench, we first used 4 cm diameter mirrors. A typical profile is shown in figure 3. 4.2. The transfer function The measurement procedure is independent of the beam diameter and of the sampling interval. It is possible to select a frequency domain by choosing particular values of w and 1x. The large spatial frequency range to which we have access is one characteristic of the method. It is possible to optimize 1x in order to minimize aliasing. This optimal 1x enables us to plot the transfer function between the real and measured slope, which shows the large spatial frequency range of the instrument. If we take into account the sampling and the laser spot size w = 250 µm, we have " # 2 1/2 2x 2 ∂z ∗ exp − 2 × III1x S(i) = ∂x x=xi πw2 w (18) where * denotes a convolution and III1x is the comb function. In Fourier space, π 2 ν 2 w2 ∗ III1x −1 (19) S(ν) = 2iπ νζ (ν) exp − 2 where ζ (ν) is the z(x) Fourier transform. The sampling interval 1x can be chosen as small as 10 µm, limited by the translation stage motor step. To avoid aliasing we choose 1x so that the exponential part in (19) becomes negligible when ν = 1/(21x). The criterion that we choose is π 2 w2 (20) ≤ e−2 exp − 2(21x)2 1036

so that 1x ≤ π w/4.

(21)

The transfer function elementary pattern is thus S(ν) π 2 ν 2 w2 T (ν) ≡ = exp − . 2iπ νζ (ν) 2

(22)

For practical reasons we will choose 1x = 200 µm and w = 250 µm. The accessible spatial frequency range is [1/D, 1/(21x)], but the sensitivity is low at low frequency, as can be seen from equation (22) S(ν) → 2iπ νζ (ν); for ν→0

this reason the bench is most sensitive to spatial frequencies near 1 mm−1 , which is very useful for testing surfaces down to the millimetre scale. 4.3. The sensitivity The altitude profile sensitivity is defined as the profile z˜ (x) of an ideally flat mirror. This ideally flat mirror is simulated by keeping the sample stationary. In that case the slope measured should always be the same and the profile would be constant and equal to zero if we use (15) and (16). Because it is not, the profile obtained can be considered as the instrument noise, namely the sensitivity. As shown in figure 4, the sensitivity is of the order of 0.1 nm for a 4 cm profile. Using equation (22) one can see that the sensitivity ˚ in the millimetre range. increases to better than 3 A 4.4. The reproducibility Reproducibility is also a fundamental parameter but it is harder to attain high reproducibility than it is to obtain


high sensitivity because reproducibility is influenced by some external phenomena: it depends on the bench thermal environment and is influenced most by thermomechanical drifts of the various components. We estimate the bench reproducibility by comparing two profiles measured in a 24 h interval. The reproducibility curve has the shape shown in figure 5. One can see that the measurement of the worst point can be repeated within a 10 nm range in 24 h. 4.5. The profile standard deviation We can estimate the profile standard deviation by conventional methods to compare it with sensitivity and reproducibility. An estimation can be made using (17): 1x 1x ∂z − (j − 1) z˜ (j ) = ∂x x=x1 2 N q=j X−1 ∂z 1x + (j − 1) 1x − ∂x x=xq N q=2 ∂z 1x 1x + − (j − 1) ∂x x=xj 2 N q=N X ∂z 1x (j − 1) (23) − + ∂x x=xq N q=j +1 so, using the standard formula, in which σ ((∂z/∂x)|x=xi ) denotes the measurement standard deviation at point xi , we have 2 1x 1x g)) = σ 2 ∂z − (j − 1) 6 2 (z(j ∂x x=x1 2 N 2 q=j −1 X ∂z 1x (j − 1) + σ2 1x − ∂x x=xq N q=2 2 1x ∂z 1x − (j − 1) +σ 2 ∂x x=xj 2 N 2 q=N X ∂z 1x (j − 1) + σ2 . (24) − ∂x x=xq N q=j +1 We hypothesize that all points are measured with the same standard deviation ∂z σ =σ ∂x x=xq and thus g)) = 1xσ 6(z(j

−

1 3 + j − (j − 1)2 2 N

1/2 .

(25)

The point of a profile which is known with the worst 6 is the central one for which j = N/2 − 1 if N is even. In that case we have

N 1 N 1/2 6 z˜ −1 = 1xσ − + 2 2 4 Dσ 1x √ σ N= √ . ' 2 2 N

(26)

σ was found experimentally to be close to 40 nrad. The worst 6 value for a 4 cm diameter sample is 0.06 nm. 5. Conclusion An instrument based on local slope measurement at a 1064 nm wavelength with no need for a reference surface has been realized. This instrument has been used to measure profiles of a 4 cm diameter mirror. It has shown a sensitivity of 0.2 nm over a large spatial frequency range; however, its reproducibility over 24 h is 4 nm. The measurement procedure is independent of the beam radius and of the sampling interval, so any spatial frequency domain can be investigated and the maximum profile length is only limited by the translation stage travel. The Virgo measurement specifications have been attained for a mirror profile and it has been shown that the measurement principle is suitable for our task. The instrument development will now continue with the addition of a rotation stage to support the sample, so that whole surfaces can be measured. References [1] Bradaschia C and del Fabbro R 1990 The Virgo Project – a wide band antenna for gravitational-wave detection Nucl. Instrum. Methods A 289 518–25 [2] Hello P and Vinet J Y 1993 Phase noise induced by fluctuations of absorbed power in mirrors, Virgo note PJT93/02 unpublished, available on request [3] 1995 VIRGO Final Design, 4300.2–4300.20 [4] Bruce C F and Sharples F P 1975 Relative flatness measurements of uncoated optical flats Appl. Opt. 14 3082–5 [5] Schulz G and Schwider J 1976 Interferometric testing of smooth surfaces Progress in Optics XII ed E Wolf (Amsterdam: North-Holland) [6] Fritz B S 1984 Absolute calibration of an optical flat Opt. Eng. 23 379–83 [7] Schulz G 1993 Absolute flatness testing by an extended rotation method using two angles of rotation Appl. Opt. 32 1055–9 [8] Schulz G and Grznna J 1992 Absolute flatness testing by the rotation method with optimal measuring-error compensation Appl. Opt. 31 3767–80 [9] Schulz G, Schwider J, Hiller C and Kicker B 1971 Establishing an optical flatness standard Appl. Opt. 10 929–34 [10] Feder J 1988 Fractals (New York: Plenum) [11] Dumas P, Bouffakhreddine B, Amra C, Vatel O, Andr´e E, Galindo R and Salvan F 1993 Quantitative microroughness analysis down to the nanometer scale Europhys. Lett. 22 717–22 [12] Loriette V and Gleyzes P 1995 Virgo ’97 static 1D simulation, Virgo note NTS029 unpublished, available on request

1037