A new analysis for diffraction correction in optical interferometry

Dimensional measurements by laser interferometry require a correction because of diffraction, which makes the fringe period different from the wavelength of a plane wave. The fractional correction—from parts in 10−7 to parts in 10−9, depending on the beam collimation—is half the central second moment of the angular power-spectrum of the beam, a generalization of the divergence concept. We report new insights into the second moment measurement and their consequences on the measurement of the silicon lattice parameter by combined x-ray and optical interferometry.


Introduction
The dispersion relation λ = c/ν is valid only for a plane wave. In a laser beam, since the wavefronts bend, the wavelength is different from one point to an other and it is different from c/ν. This effect it is known as diffraction and is an unavoidable phenomenon connected with the wave nature of light.
Diffraction has been extensively investigated to correct dimensional measurements by laser interferometry [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. When the interfering beams are identical, the detector is infinite, the optical path difference is small, and the displacement to be measured occurs along the beam axis, the fractional difference of the fringe period from the plane wave wavelength is proportional to the divergence square which, for generic paraxial beams, is expressed by the eigenvalues of the secondmoment matrix of the angular power-spectrum.
For the dissemination of the unit of length, lasers are calibrated against frequency standards. The frequency calibration achieves relative uncertainties smaller than 10 −10 , but translating frequency into wavelength requires the characterization of the laser beam and the correction for the effects of diffraction. By using Gaussian approximations of the interfering beams, the correction is proportional to the divergence square and, typically, ranges from parts in 10 −7 to parts in 10 −9 . However, high accuracy measurements and poor beam quality require the determination of the second central moment of the angular power-spectrum, which is the way to generalize the divergence concept to arbitrary paraxial beams. In this paper, we review the angular power spectrum measurement and the basic formulae for the correction calculation and illustrate a new view onto the data analysis, which exploits symmetries to filter the noise without loss of information and to gain visibility of the beam diffraction.
Next, our new view is applied to the INRIM's measurements of the lattice parameter by combined x-ray and optical interferometry [15][16][17]. These measurements were corrected by imaging the angular power-spectrum in the focal plane of a converging lens [18]. The required accuracy raised questions about the power-spectrum measurements and correction estimates. Recently, a joint INRIM and PTB investigation brought into light the presence of small spectrum components at relatively large angles-consistent with the Airy patterns from diffraction-limited fiber-collimator lensesthat went unnoticed in the previous analyses. Section 4 compares the present and previous data analyses and shows the improvement achieved. The updated corrections are given in section 5.

Fourier optics
The angular spectrum of a paraxial beam, u(x, y; z)e ikz , which propagates along the z axis with wave-number k, is its Fourier transform in terms of plane waves with variable propagation directions [5,19]. In (1), x,y are the transverse coordinates, p = kθ x and q = kθ y are the reciprocal coordinates, and the θ x,y angles identify the propagation directions of the plane waves. The beam axis, i.e. the propagation direction of the u(x,y;z) center of mass, π 0 , is identified by the center of mass of the power spectrum |ũ( p, q)| 2 . Hence, where π = ( p, q) T , the power spectrum is independent of the z coordinate, and we assumed that u(x,y;z) and ũ( p, q; z) have unit norm. The second central moment is (3) and it generalizes the divergence concept.

Diffraction correction
In the interference of two identical and coaxial paraxial beams, whose complex amplitudes differ only by free-space propagation, the difference between the period of the interference pattern integrated by an infinite detector, λ e , and the plane-wave wavelength, λ = 2πc/ω, is where the propagation difference is assumed much smaller than the Rayleigh length and γ 2 1,2 are the eigenvalues of the second central moment matrix Γ 0 of the beam angular power spectrum. Equation (4) is the weighted average of each ray cosineerror, it originates from the spread of the transverse impulse of the photons, and it holds for any paraxial beam, no matter if its profile is Gaussian or not. In the case of Gaussian beams having cylindrical symmetry, ∆λ/λ = θ 2 0 /4 = 1/(kw 0 ) 2 , where ∆λ = λ e − λ, and θ 0 and w 0 = 2/(kθ 0 ) are the beam divergence and the 1/e 2 radius at the waist.
If the interfering beams are Gaussian, the generalization of (4) to non-coaxial beams, is given in [7]. In (5), x 0 is the shear of the beam axes at the beam waists, 2α is the misalignment between the beam axes, and β is the angular misalignment between the measurand displacement and the axis of the measurement beam. The β 2 /2 term is of geometrical origin; it expresses that the displacement occurs at an angle β with respect to the beam axis. The α 2 /2 and x 0 /w 0 terms take the decoupling of the interfering rays, which is due the combination of displacement and spread of the ray bundle, into account. In order to calculate ∆λ/λ by the focal plane method [18], a key observation is that γ 2 1 + γ 2 2 = Tr(Γ 0 ), where Tr(Γ 0 ) is the trace of second moment matrix. Hence, (4) can be rewritten as A second key observation is that the calculation of Tr(Γ 0 ) requires the integration of the univariate function I(θ) resulting from the power-spectrum averaging over the azimuth φ of the propagation directions. In fact, where I(θ) = |ũ(kθ)| 2 is the normalized power-spectrum, θ = (θ x , θ y ) is any propagation direction, and θ = |θ| = θ 2 x + θ 2 y and φ = arctan(θ y /θ x ) are elevation and azimuth coordinates in the reciprocal space, whose origin is the beamaxis direction. Equation (7) makes it possible to reduce the noise of the focal plane image, to trace back the identification and elimination of the camera dark-noise to a more tractable one-dimensional fitting problem, and to bring into light spectrum-details otherwise invisible.

Combined x-ray and optical interferometry
An x-ray interferometer consists of three Si crystals cut so that the {2 2 0} diffracting planes are orthogonal to the crystal surfaces [15,16]. X rays (17 keV Mo K α source) are split by the first crystal and recombined by the third, which is called the analyser, via a mirror crystal. When the analyser is moved orthogonally to the {2 2 0} planes, a periodic variation of the transmitted and diffracted x-ray intensities is observed, the period being the diffracting-plane spacing, d. The analyser embeds a front mirror, so that its displacement is measured by optical interferometry. The measurement equation is d = mλ/(2n) where n is the number of x-ray fringes in m optical fringes of λ/2 period.
In 2014, the INRIM made seven measurements of the lattice parameter. Three from February to March, with the interferometer crystals arranged as in the parent ingot (side A). Four from May to June, with the inverted analyser (side B). The measurements, whose accuracy approached 10 −9 d, were corrected for the difference between the effective period of the integrated signal and the wavelength of a plane wave. In each measurement campaigns, the angular power-spectrum of the beam emerging from the optical interferometer was measured by using the Fourier transforming properties of a lens. Exemplar results are shown in figure 1.
In February, we used a Hilger and Watts TA3 autocollimator equipped with a COHU 6710 CCD camera; the focal plane calibration was 7.2(2) µrad/pixel, but the resolution was limited to 8 bit and the captured images were affected by a large dark noise. The fixed-pattern noise visible in the residuals is due to non-uniformity of the dark currents on the chip; the dark spots are due to dust particles inside the autocollimator.
In May, we used a converging lens having 0.5 m focal length; the focal plane image was acquired by a cooled 16 bit HAMAMATSU CMOS camera FLASH 4.0 having much less readout and dark noises. The focal plane was identified by checking that parallel translations of a laser beam do not cause translations of the beam image; calibration, about 13.3(4) µrad/pixel, was accomplished by tilting the beam by known angles. The linear fringe pattern in the residuals-also visible in the February data-might be due to a scratch on the interferometer optics. The rotation between the February and May images is due to the different assembling of the cameras.

Data analysis
Up to now, we did not realize that, since it can be expressed in term of Tr(Γ 0 ) (which is independent of the Γ 0 representation), ∆λ/λ depends only on the second moment about the center of mass and it can be obtained by the application of (7). Therefore, we resorted to the Γ 0 eigenvalues and estimated the full second-moment matrix of the images. Since the dark noise limits the accuracy of the Γ 0 calculation by sums over the pixels (the noise in the tails has the heaviest weight and errors in identifying and subtracting it yield oversized or undersized estimates), the pixels far from the center of mass can be advantageously discarded [20]. Attention was given to the trade-off between clipping and noise rejection; but, as part of the spectrum is rejected too, there were unavoidable subjectivity and uncertainty. Therefore, in [15,16], we estimated Γ 0 by fitting the bivariate Gauss function to the focal plane data I FP (θ i ). The results are shown in figures 1 and 2.
The PTB developed a procedure to obtain the angular spectrum via the discrete Fourier transform of two-dimensional samples of the complex amplitude estimated from wavefront and intensity profiling by means of a Shack-Hartmann sensor [15]. In this case, the intensity and wavefront samples are fitted by using bivariate Gauss functions (intensity samples) and Zernike polynomials (wavefront samples) as bases. Next, the results are combined to give the complex amplitude, which is re-sampled to calculate the angular spectrum by discrete Fourier transform. Owing to the data smoothing, the spectrum is free of noise, so that the mass center and second moments can be obtained from sums over the samples.
A comparison of the INRIM and PTB power-spectrum measurements, done by means of travelling standards realized in the form of single-mode fibers equipped with collimators, brought into evidence that the INRIM's smoothing of the focal-plane images by Gauss functions was unable to take small spectrum components at relatively large angles into account. As shown in figure 2 and [16], this problem went unnoticed, but it is brought into light by the logarithmic plots shown in figures 3 and 4.
In order to eliminate the difficulty of capturing these comp onents by fitting analytically-specified freeform shapes to the two-dimensional images, the PTB rethought about the problem and realized that it can be made one-dimensional as shown in (7). Since the impact of noise is reduced in the processing, I(θ) can be approximated binning the pixels having the same (rounded) radial coordinate. The integration over the radial direction is done by fittingÎ where the Gauss-Laguerre functions are used as basis, to the binned data, I FP (θ i ). When the spectrum components to be modelled are many orders of magnitude smaller than the spectrum peak, it is convenient to fit Figures 3 and 4 show the binned data and compares (9a) and (9b) against Gauss functions. It is worth noting that (9a) and (9b) are tools to remove the dark noise, to smooth the binned data and to ensure that Î (θ) > 0 and that the integrals in the trace calculation converge and are insensitive to the dark noise. From a physical viewpoint, (9a) and (9b) model effective beams having the same central second-moment, but cylindrical symmetry. Figure 5 shows the  The initial estimate of the beam-axis direction is found by the fitting of a bivariate Gauss function to the data. Since the trace of the non-central second-moment matrices, where θ 1 is any direction, exceed Tr(Γ 0 ) by the square of the θ 1 distance from the beam axis direction θ 0 , the center uncertainty propagates into an overestimate. It is worth noting that β 1 = |θ 1 − θ 0 | is the angular misalignment between the beam axis and the θ 1 direction. Therefore, if the displacement occurs along θ 1 , the Tr(Γ 1 )/2 error includes the β 2 1 /2 contribution of (5). This brings into light that the splitting of the fringe-period error in diffraction and cosine contributions is conventional. Equation (10) suggests to improve the correction estimate by minimizing the trace calculation versus the non-centrality of the second moment. As it is shown in figure 6, this is done by fitting the parabola to the traces of a number of non-central moments.

Results
The  In 2010, the measured values of the lattice parameter carried out with the analyser assembled in the front (side A) and inverted (side B) layouts were separately corrected; in 2014, all the measured values were corrected for the same, average, error. The new corrections values have been obtained by reexamining the archive files of the angular power-spectrum measurements and by recalculating the second central moments. All the corrections were obtained by minimizing the trace calculation versus the non-centrality of the second moment.
Four terms contribute to the correction uncertainty: the calibration of the focal plane coordinates, the focus of the autocollimator and telescope, the integration of the noisy data points, and scattering by dust particles in the beam path.      The uncertainty of the focal plane calibration is 4%. The fractional error due to a detection offset σ z ≈ 2 mm with respect to the focal plane is where F = 0.5 m is the focal length and z 0 the distance of the beam waist from the lens principal plane. We don't know were exactly is the beam waist, but assuming |F − z 0 | ≈ 3 m, which is compatible with the size of the experimental set-up, the fractional uncertainty is 5%. As regards the momentum calculation, the observed variability associated to different binning of the power-spectrum samples and the number of basis functions is about 5%. Though this uncertainty is reduced by the averaging over different binnings implied by the momentum minimization, we nevertheless maintained the 5% fractional uncertainty. Eventually, the last source of uncertainty is the forward scattering by dust particles. Putting it all together, we cautiously associated to the 2017 corrections given in table 1 a fractional uncertainty equal to 15%.
In [16], to check the correction estimate, we examined the results of a number of test d 220 measurements carried out from 2010 to 2014 with different laser beams. The measurement results are shown in figure 7 together with the relevant correction for the diffraction. The corrected values given in [16] and reproduced in figure 8 (left) suggested overestimated corrections, but we were unable to shed light on the problem. All the archive files of the angular powerspectrum measurements made at the time were reexamined and the second central moments recalculated by fitting (9a) and (9b) to the focal plane data. Though these test measurements were not explicitly carried out to study diffraction and not all of them own a full experimental check of the error budget, the results given in figures 7 and 8 (right) show that the supposed overestimate disappears and confirm the improvement achieved.

Conclusions
An error was found in the estimate of the correction for the laser-beam diffraction of the INRIM measurements of the 28 Si lattice parameter by combined x-ray and optical interferometry [15,16]. Specifically, the correction-which is proportional to the second central moment of the angular power-spectrumwas calculated as relevant to the bivariate Gaussian-beam that most closely approaches the measured spectrum. However, small spectrum components at large angles went unnoticed, but they had a significant effect on the momentum.
The archived measurements of the angular power-spectrum were reexamined and the second central moments recalculated by exploiting symmetries to filter the noise and to reduce the calculation to a one-dimensional integration. This allowed for high-order analytical approximations of the one-dimensional data and more accurate momentum calculations. To allow the lattice parameter values given in [15,16] to be re-corrected, the newly calculated corrections have been given, together with the applied ones.
The results obtained bring into light the importance of beam characterization when calibrating wavelength standards for optical interferometry. The transposition of the frequency uncertainty into the wavelength one might lead to underestimations; high-accuracy measurements or poor beam quality might also require the specification of the beam angular spectrum.