Wavefront errors in a two-beam interferometer
WWavefront errors in a two-beam interferometer
G Mana, E Massa and C P Sasso
INRIM - Istituto Nazionale di Ricerca Metrologica, Str. delle Cacce 91, 10135Torino, ItalyE-mail: [email protected]
Abstract.
The paper deals with the impact of wavefront errors, due to theoptical aberrations of a two-beam interferometer, on the period of the travellingfringe observed by integrating the interference pattern. A Monte Carlo simulationof the interferometer operation showed that the fringe-period estimate is unbiasedif evaluated on the basis of the angular spectrum of the beam entering theinterferometer, but the wavefront errors increase the uncertainty.
Submitted to:
Metrologia
PACS numbers: 42.25.Fx, 06.30.Bp, 07.60.Ly
1. Introduction
In length metrology by optical interferometry, the wavefront errors affect the period ofthe interference signal. The calibration of lasers against frequency standards achievesrelative uncertainties smaller than 10 − , but it is not possible to trace back thewavelength to the frequency via the plane-wave dispersion equation. The relevantcorrections have been extensively investigated in the literature [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12, 13, 14]. When the interfering wavefronts differ only by the propagationdistances through the interferometer arms, the fractional wavelength difference –which, typically, ranges from parts in 10 − to parts in 10 − – is proportional to thesquare of the beam divergence which, for arbitrary paraxial beams, is proportional tothe trace of the second central-moment of the angular power-spectrum [15, 16].Characterizations of the laser beams leaving a combined x-ray and opticalinterferometer brought into light wavefront and wavelength ripples having a spatialbandwidth of a few mm − and amplitudes as large as ±
20 nm [17] and ± − λ [18], respectively, which might have a detrimental effect on the accuracy of themeasurements. Since the differential wavefront-errors – i.e., a non-uniform phaseprofile of the interference pattern – cannot be explained by aberrations of beam feedingthe interferometer, we carried out an analysis of the effect of wavefront aberrationsdue to the interferometer optics.In section 2, we outline the mathematical framework needed to model two-beam interferometry and paraxial propagation and show a one-dimensional analyticalcalculation of the difference of the fringe period from the plane-wave wavelength.Eventually, we report about a Monte Carlo two-dimensional calculation of the fringeperiod in the presence of wavefront errors caused by the interferometer optics. a r X i v : . [ phy s i c s . i n s - d e t ] J u l
2. Mathematical model
The interferometer slides two beams, u ( r ; z + s ) exp( − i kz ) and u ( r ; z ) exp( − i kz ),one with respect to the other by a distance s while keeping them coaxial. By leavingout the exp( − i kz ) term of the optical fields – where k = 2 π/λ is the plane-wave wavenumber and z the propagation distance – and assuming and infinite detector, theintegrated interference signal is S ( s ) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ | u ( r ; s ) + u ( r ; 0) | d r = (cid:90) + ∞−∞ (cid:90) + ∞−∞ | ˜ u ( p ; s ) + ˜ u ( p ; 0) | d p , (1)where we reset the origin of the z axis, r is a position vector in the detector plane(orthogonal to the z axis), ˜ u ( p ; s ) and ˜ u ( p ; 0) are the angular spectra of theinterfering beams [19], and p is the wavevector of the angular spectra basis, exp( − i pr ).The phase of the integrated interference pattern in excess (or defect) with respectto − ks is [15] Φ( s ) = arg (cid:2) Ξ( s ) (cid:3) , (2)where Ξ( s ) = (cid:90) + ∞−∞ ˜ u ∗ ( p ; 0) U ( p ; s )˜ u ( p ; 0) d p (3)is the interference term of the integrated intensity. In (3), we used ˜ u ( p ; s ) = U ( p ; s )˜ u ( p ; 0), where U ( p ; s ) = exp (cid:18) i p s k (cid:19) , (4)is the reciprocal space representations of the paraxial approximation of the free-spacepropagator and p = | p | . The fringe period is λ e = λ (cid:18) k dΦd s (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:19) , (5)where the sign of the derivative is dictated by the negative sign chosen for the plane-wave propagation. It must be noted that, since U ∗ ( p ; z ) U ( p ; z + s ) = U ( p ; s ), theinterfering beams can be propagated by the same distance z without changing (3)and, consequently, λ e . Therefore, (5) depends only on the length difference of theinterferometer arms, not on the detection-plane distance from, for instance, the beamwaist.The interferometer recombines the light beams after delivering it through armsof different optical lengths. We consider the case when the interferometer arms havethe same length; an analysis of the fringe phase and period as a function of thearm difference is given in [6]. However, we want to allow the interferometer arms todeviate from perfection. Therefore, ˜ u ( p ; 0) and ˜ u ( p ; 0) are intrinsically different,meaning that they cannot be made equal by freely propagating one of the two, and,as implicitly assumed in (3), the aberrations occur after the beam splitting but beforethe interferometer mirrors. To give an analytical one-dimensional example, let the complex amplitudes of thedirect space representation of the interfering beams differ by a small wavefront error ϕ ( x ), that is, u ( x ) = u ( x )e i ϕ ( x ) ≈ u ( x ) (cid:2) ϕ ( x ) − ϕ ( x ) / (cid:3) , (6)where we omitted the z = 0 specification, and let u ( x ) = (cid:18) πw (cid:19) / e − x /w (7)be a normalized Gaussian beam. Since we are interested to small sliding distance withrespect to the Rayleigh length – that is, ksθ / (cid:28)
1, where θ is the beam divergence– it is convenient to use a finite difference approximation of the z derivative in theparaxial wave equation and the first-order approximation, U ( x ; s ) ≈ − i s∂ x k , (8)of the direct-space representation of the free-space propagator. Hence,Ξ( s ) = (cid:90) + ∞−∞ u ∗ ( x ) (cid:18) − i s∂ x k (cid:19) u ( x ) d x. (9)It is worth noting that, since the − ∂ x operator is self-adjoint, it does not matter what ofthe interfering beams is slid. Therefore, for the convenience of the Ξ( s )’s computation,we choose to propagate u ( x ). By using (6) and carrying out the integrations in (9),we obtain [21]Re[Ξ( s )] = √ π − skw (cid:90) + ∞−∞ e − ξ (1 − ξ ) ϕ ( ξ ) d ξ − (cid:90) + ∞−∞ e − ξ ϕ ( ξ ) d ξ (10 a )andIm[Ξ( s )] = √ πs kw − s kw (cid:90) + ∞−∞ e − ξ (1 − ξ ) ϕ ( ξ ) d ξ + (cid:90) + ∞−∞ e − ξ ϕ ( ξ ) d ξ, (10 b )where ξ = √ x/w . The plane-wave wavelength is shorter than the fringe period λ e as defined in (5); thefractional difference is [21]∆ λλ ≈ θ (cid:20) π (cid:90) + ∞−∞ e − ξ (1 − ξ ) ϕ ( ξ ) d ξ (cid:90) + ∞−∞ e − ξ ϕ ( ξ ) d ξ − √ π (cid:90) + ∞−∞ e − ξ (1 − ξ ) ϕ ( ξ ) d ξ (cid:21) , (11)where the calculation was carried out up to the second perturbative order, ∆ λ = λ e − λ , θ = 2 / ( kw ) is the u ’s divergence, and ξ = √ x/w .The simplest way to investigate the effect of the wavefront ripple is to considerthe phase grating ϕ ( ξ ) = (cid:15) sin( aξ + α ) , (12) i = = - - - - - - w / Λ p r opaga t i ond i r e c t i on / n r ad i = = = - w / Λ de l t a v a l ue / % Figure 1.
Left: propagation directions of the aberrated and the superimposedbeams ( u and u + u , respectively) exiting the interferometer. Right: deltavalues of the approximate wavelength differences (14 a - c ) relative to the averagedifference (16) vs. the fractional spatial frequency w / Λ of the wavefront error(12). The root-mean-square amplitude of the wavefront error is 10 nm. where a = 2 πw / ( √ (cid:15) (cid:28) λλ ≈ θ (cid:34) a e − a cos(2 α ) + (2 + a )e − a / sin ( α )2 (cid:15) (cid:35) (13)The θ / u ( x ) angular spectrum. Itis the one-dimensional equivalent of half the trace of the second central-moment of theangular spectrum [15, 16], which is the standard ingredient to calculate the neededcorrection and takes the diffraction of arbitrary paraxial beams into account.To quantify the impact of the wavefront error, we compare the fractional difference(13) to the approximations ∆ λ/λ ≈ Tr( Γ i ) /
2, where Γ i is: i) the second central-moment of the angular spectrum of the unperturbed beam u illuminating theinterferometer, ii) the aberrated, u , and iii) superimposed, u + u , beams exitingthe interferometer. In the first case we haveTr( Γ ) / θ / , (14 a )in the second Tr( Γ ) / θ (cid:104) a (1 + e − a − − a / ) (cid:15) (cid:105) , (14 b )and, in the third,Tr( Γ ) / θ (cid:20) a (1 + 2e − a − − a / ) (cid:15) (cid:21) . (14 c )In (14 b - c ), we used the approximation (6) and, for the sake of simplicity, set α = 0. It isworth noting that, as shown in Fig. 1 (left), the propagation directions of the aberratedand superimposed beams, u and u + u , deviate from that of the unperturbed beam u by θ = − a e − a / (cid:15)/k and θ = − a e − a / (cid:15)/ (2 k ), respectively. The misalignmentoccurring when Λ /w ≈ a - c ), δ i = Tr(Γ i ) / − ∆ λ/λ ∆ λ/λ , (15)relative to the fractional difference (13) evaluated with α = 0,∆ λ/λ = θ (cid:32) a e − a (cid:15) (cid:33) , (16)are shown in Fig. 1 (right). In the case of a phase-grating pitch equal or shorter thanthe beam diameter, the increased angular spread of the aberrated beam does not affectthe fringe period. We do not have an explanation of this.
3. Numerical analysis
The analytical treatment of section 2.3 suggests that the actual fringe period mightbe different from the estimate based on the second central moment of the angularspectrum. Since the oversimplified analysis can hardly quantify this difference andthe associated uncertainty, we resorted to a Monte Carlo estimate.In the simulation, the two interfering beams, u i ( x, y ) = (cid:2) A i ( x, y ) / (cid:3) g ( x, y )e i ϕ i ( x,y ) , (17)were independently generated 10 times. In (17), A i ( x, y ) and ϕ i ( x, y ) are the intensityand phase noises and g ( x, y ) = e − ( x + y ) /w (18)is the Gaussian beam feeding the interferometer, where w = √ A i ( x, y ) and ϕ i ( x, y ) werecollections of Gaussian, independent, and zero-mean random variables indexed by theobservation-plane coordinates. As shown in Figs. 2 and 3, they have σ ϕ = 10 nm and σ A = 0 .
025 standard deviations and were filtered so has to have the same correlationlength of about 0.5 mm observed experimentally [17, 18]. We did not consider thewavefront curvature and imperfect recombinations of the interfering beams, whichmight be modelled by amplitude and phase perturbations [6].The Monte Carlo simulation proceeded by Fourier transforming u ( x, y ) and u ( x, y ) and by calculating their interference and excess phase according to (3) and(2). Figure 4 shows the angular spectrum of the aberrated beam. The plateau at 10 − mrad − , which extends up to about 1 mrad, originates from the A ( x, y ) and ϕ ( x, y )noises. According to (5), the Monte Carlo values of the fractional difference betweenthe fringe period and λ ,∆ λλ (cid:12)(cid:12)(cid:12)(cid:12) MC = ∆Φ2 πs/λ , (19)where ∆ λ = ( λ e − λ ) /λ , are obtained by propagating the fields back and forward by s/ ± λ and by calculating the phase difference∆Φ = arg[Ξ( s/ − arg[Ξ( − s/ , (20)which is null when the fringe period is equal to the plane-wave wavelength λ .To check the numerical calculations, we considered some one-dimensional cases,where the analytical expressions of the difference between the fringe period and theplane-wave wavelength are available, and compared the numerical calculations againstthe values predicted by (11). The results are summarized in table 1. - - - - - - - - x / mm y / mm - - - - - - - - x / mm y / mm Figure 2.
Left: simulated wavefront error; the colour scale spans ±
30 nm. Right:residuals from a Gaussian of the simulated intensity profile; the colour scale spans ±
2% of the maximum beam intensity. The standard deviations of the wavefronterrors and intensity profile are σ ϕ = 10 nm and σ A = 0 . - - - - - - x / mm w a v e f r on t e rr o r / n m i n t en s i t y / a r b i t r a r y un i t s Figure 3.
Orange: y = 0 section of the differential wavefront error shown in Fig.2. The blue line is the same section of the (aberrated) intensity profile. Table 1.
Comparison of analytical, equation (11), and numerical, equation (19),calculations of the fractional difference (expressed in nm/m) between the fringeperiod and the plane-wave wavelength in some one dimensional cases. case σ A σ A σ φ /nm σ φ /nm Eq. (11) numerics A = A = 0 , ϕ = ϕ = 0 − − − − A = A (cid:54) = 0 , ϕ = ϕ (cid:54) = 0 0.025 0.025 50 50 4.762 4.762 A (cid:54) = 0 , A = 0 , ϕ (cid:54) = 0 , ϕ = 0 0.025 − − A = 0 , A (cid:54) = 0 , ϕ = 0 , ϕ (cid:54) = 0 − −
10 1.719 1.724 - - - - - - - - θ x / mrad θ y / m r ad - - θ / mrad angu l a r po w e r s pe c t r u m / a r b i t r a r y un i t s Figure 4.
Left: residuals of the angular power spectrum after subtracting thebest-fit spectrum of a Gaussian beam; the colours indicate the normalized density.Right: averaged radial plot of the angular power spectrum (orange dots); the bluedots are the angular spectrum of the Gaussian beam feeding the interferometer.The standard deviations of the wavefront error and intensity profile are σ ϕ = 10nm and σ A = 0 . To quantify the effect of two-dimensional wavefront errors, the fractionaldifferences numerically calculated were compared to the approximations,∆ λλ (cid:12)(cid:12)(cid:12)(cid:12) i = 12 Tr( Γ i ) , (21)which holds when the interferometer does not aberrate the interfering beams [15, 16],used to correct the interferometric measurements in [22, 23]. In (21), Γ i is thesecond central-moment of the angular spectrum of: 1) the Gaussian beam feeding theinterferometer, | ˜ g ( p ) | , 2) the interfering beams leaving the interferometer, | ˜ u ( p ) | and | ˜ u ( p ) | , and 3) the interfering-beam superposition, | ˜ u ( p ) + ˜ u ( p ) | . Thefractional delta values of the approximate differences (21) relative to the numericalone (19), δ i = Tr(Γ i ) / − (∆ λ/λ ) MC (∆ λ/λ ) MC , (22)are shown in Fig. 5.The difference estimated from the angular spectrum of the beam entering theinterferometer is equal to the mean of the actual (numerically calculated) values, whichare scattered by about 12%. Contrary, the differences estimated from the angularspectra of the beams leaving the interferometer, superposed or not, are significantlylarger than truth. It is worth noting that the period values separately calculated fromthe angular spectra of each of the two beams leaving the interferometer are statisticallyidentical. This agrees with the observation that the Monte Carlo simulation does notdistinguish between the interfering beams. - -
20 0 20 40 60 80 1000.000.010.020.030.04 delta value / % p r obab ili t y den s i t y mean = = % - -
20 0 20 40 60 80 1000.000.020.040.060.080.10 delta value / % p r obab ili t y den s i t y mean = % standard deviation = %- -
20 0 20 40 60 80 1000.000.020.040.060.080.10 delta value / % p r obab ili t y den s i t y mean = % standard deviation = % - -
20 0 20 40 60 80 1000.000.020.040.060.080.10 delta value / % p r obab ili t y den s i t y mean = % standard deviation = % Figure 5.
Distributions of the delta values of the approximate wavelengthdifferences (21) – obtained from the angular spectra of: the Gaussian beam feedingthe interferometer, | ˜ g ( p ) | (case 1, top left); the interfering-beam superposition, | ˜ u ( p )+˜ u ( p ) | , (case 3, top right); the beams leaving the interferometer, | ˜ u ( p ) | and | ˜ u ( p ) | , (case 2, bottom left and right) – relative to the numerical difference(19). The standard deviations of the wavefront errors and intensity profile are σ ϕ = 10 nm and σ A = 0 .
4. Conclusions
We observed that the laser beams leaving the combined x-ray and opticalinterferometer used to measure the lattice parameter of silicon display wavelength andphase imprints having a spatial bandwidth of a few mm − and local wavefront errorsand wavelength variations as large as ±
20 nm and ± − λ [18]. These aberrations arelikely due to the interferometer optics. Besides, the observed imprints correspond toa root-mean-square deviation from flatness of each of the optics’ surfaces of less than3 nm for scale lengths from 0.1 mm to 2 mm.Since our measurements, which were corrected on the basis of the angular spectraof the laser beam, aimed at 10 − fractional accuracy, questions arise about the impactof these errors. The Monte Carlo simulation of the interferometer operation indicatesthat the corrections made depend on the angular spectra having been measured beforeor after the interferometer. The correction is faithfully evaluated when the spectrumis measured before the interferometer.Unfortunately, not being aware of the problem, we measured the angular spectraafter the interferometer [24, 25, 16]. However, we note that the excess of correction isdue to the angular-spectrum plateau that increases the central second-moment of theincoming beam. Since, as shown in Fig. 4 (right), this plateau is indistinguishable fromthe instrumental background of the spectrum measurement, it was subtracted from thedata and excluded from consideration. Therefore, the plateau and, consequently, thewavefront errors did not bias the corrections made. By using the typical aberrationsobserved in our set-up, the correction uncertainty is 12%, which is within the 15%cautiously associated with them [16].Nevertheless this reassuring conclusion, our work evidenced unexpected criticalissues, which deserve further investigations and on-line determinations of the neededcorrection, e.g., by reconstructing its value from the (measurable) dependence on thedetector area. These results have a value also in other experiments, such as thosedetermining the Planck constant and the local acceleration due to gravity, whereprecision length-measurements by optical interferometry play a critical role. References [1] Dorenwendt K and Boensch G 1976
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