## Abstract

Digital wave-front measuring interferometry is a well-established technique but only few investigations of systematic error sources have been carried out so far. In this work three especially serious error sources are discussed in some detail: inaccuracies of the reference phase values needed for this type of evaluation technique; disturbances due to extraneous fringes; and spatially high frequency noise on the wave fronts caused by dust particles, inhomogeneities, etc. For the first two error sources formulas of the resulting phase deviation are derived and compensation possibilities discussed and experimentally verified. To study the occurrence of wave-front irregularities caused by dust particles a model has been developed and countermeasures derived which assure sufficient regularity of contour line plots. The repeatability of the present experimental setup was better than λ/200 within the 3*σ* limits.

© 1983 Optical Society of America

## I. Introduction

During the last decade numerous publications[1]–[4] have appeared which use the wave-front measuring technique invented by Bruning *et al.*[5] This method allows measurements of wave-front deviations in the λ/100 region with repeatabilities in the 10^{−3} λ region.

Although this method is already used in industry, investigations of systematic errors are scarce.[6] Furthermore, some statements in the literature concerning systematic error sources are at least incomplete or inconsistent concerning the attainable accuracy. Only a few investigations of nonenvironmental systematic error sources have been carried out.[4] For this reason we feel justified in discussing some systematic error sources in more detail than hitherto. Our study contains theoretical as well as experimental results. For the experiments we used the interferometric setup given in Fig. 1. This is a conventional Twyman-Green interferometer equipped with a piezoelectric transducer to vary the reference phase controlled by a highly accurate capacitive distance meter and with a 32 × 32-element matrix as well as a CCD line array with 256 elements as a detector. The photoelectrical signal from the detector is converted into an 8-bit word which is fed into a K1520 microprocessor system. The system calculates the phase distribution according to the fundamental relations given in the literature.[5] The piezotransducer is controlled by the same microprocessor. The results of the phase measurement can be displayed graphically. In Fig. 2 the wave aberrations of a micro-objective of N.A. = 0.32 are shown where the contour line distance is λ/32. [Wave aberrations *W*(*x,y*) and the phase deviation Φ(*x,y*) which is a measured quantity are connected by the relation Φ(*x,y*) = 2*kW*(*x,y*) with *k* = (2*π*)/λ. The factor 2 results from the double passage in the Twyman-Green interferometer. Apart from the factor 2*k* the expressions for phase deviations and wave aberrations are used in the text synonymously.]

The CCD line array allows only for measurement of one central section through the 2-D wave aberration, but the lateral resolution of the CCD array is 8 times higher than with the matrix array which is essential for the error analysis. Therefore, the experimental investigations were carried out mainly with the CCD array. It will become clear from the context that the restriction to one dimension can be balanced by choosing the experimental conditions accordingly.

So far the software enables calculation of the phase deviations; elimination of linear and/or an additional quadratic term in the pupil coordinates; averaging operations; difference and summation operations of consecutive runs and other data manipulations discussed in the text; and display of the phase as contour line plot, pseudo-3-D plot, and as curve for the CCD data.

(A run comprises at least the data acquisition time for a set of values enabling the calculation of the phase to be measured.)

## II. Errors Caused by Reference Phase Deviations from Ideal

The basic equation for wave-front measuring interferometry was given by Bruning *et al.*[5]:

*φ*is the reference phase

_{r}*φ*= [(2

_{r}*mπ*)/

*R*] (

*r*− 1),

*m*is a whole number, and

*I*(

_{r}*x,y*) is the intensity distribution in the detector plane in the absence of errors.

Equation (1) assumes that the r samples are taken equidistantly and that *r* runs over whole periods of the intensity distribution. Only in this case Eq. (1) is valid. If the above assumptions are violated more complex equations will arise from the least-squares formalism underlying Eq. (1).

Here, we shall discuss weak violations of the exact conditions which may be caused by reference-phase inaccuracies or by thermal drifts and mechanical relaxation during the data acquisition time of a single run. Assuming a two-beam interference pattern we can write for the intensity *I′* as the measured quantity:

*ψ*is the actual reference phase given by the PZT reference mirror,

_{r}*I*

_{0}is the mean intensity, and

*V*is the visibility in the interference pattern.

While *ψ _{r}* is a physical quantity,

*φ*is a mathematical one stored in the computer memory. For obvious reasons we introduce an error

_{r}*ɛ*of the reference phase

_{r}*φ*:

_{r}*π*because the phase is corrected for phase discontinuities anyway.

Using the orthogonality relations of the sine and cosine functions[7] and assuming that *ɛ _{r}* is small, we can put cos

*ɛ*= 1 and sin

_{r}*ɛ*=

_{r}*ɛ*. Then the following expression can be derived:

_{r}The quantities *C* and *S* can be interpreted as Fourier coefficients of the error distribution *ɛ* = *ɛ*(*r*). In this interpretation only contributions correlating with sin2*φ _{r}*, cos2

*φ*are essential. The interpretation as Fourier coefficients is also suggested by the approximate Fourier analysis contained in the algorithm of Eq. (1). In Eq. (7) the mean value of the phase error

_{r}*x,y*) are eliminated by the least-squares algorithm for the reduction of the Φ values, so the phase disturbances stem from the Φ-dependent terms. It should be noted that not a Φ dependence but a 2Φ dependence exists. Such effects can be demonstrated by choosing an appropriate position dependence of the phase to be tested. In the 1-D case we assume Φ =

*a*+

*bx*(

*a*and

*b*are constants). In interferometry this means a fringe adjustment. If

*M*is the number of fringes the phase deviations will show a periodic disturbance with 2

*M*maxima and minima. The periodic character can be understood best by developing Eq. (7) for small Fourier coefficients

*C*and

*S*. Furthermore, the influence of the reference-phase errors in Eq. (7) diminishes with at least (1/√R) because of the randomness of

*ɛ*.

_{r}Therefore, in the case of many phase steps/period the influence of this error on the phase to be measured will be weak although noticeable with high-precision measurements. However, microcomputer limitations and interferometer drifts make high-speed measurements using only few reference-phase steps desirable.[3],[4]

Equation (1) becomes extremely symmetric and simple if only four phase steps are taken, i.e., *φ _{r}* = (

*r*− 1) (

*π*/2),

*r*= 1,…,4 and, therefore, the phase Φ can be calculated from

*ɛ*

_{2}=

*π*/16,

*ɛ*

_{3}=

*π*/8,

*ɛ*

_{4}= 3

*π*/16 and made two runs where the second run was in phase opposition by the use of a reference-phase offset

*π*/2 as follows from Eq. (7). Figure 3(a) shows the difference between two such runs. (The locus on the surface is plotted in the

*x*direction and the phase Φ in the

*y*direction.) The difference is free from wave-front deformations of the interferometric components and shows only the phase distortions caused by the reference-phase errors if the interferometer has a wedge adjustment. The oscillations have twice the amplitude of a single run.

In Fig. 3(b) a section through the intensity pattern in the detection plane is shown. The cos-type fringes correspond to the phase distortion except that the spatial frequency of the phase variations has doubled.

One could argue that in normal measuring conditions a zero fringe pattern is adjusted, but it is obvious that phase variations due to the existing wave aberrations occur and therefore distortions are inevitable.

The averaging quality of the phase measurement procedure due to Eq. (1) can be studied by a comparison of runs with considerably different reference-phase step numbers. In Fig. 4 two deviation pictures are given made with the difference technique used in Fig. 3(a) but with only one arbitrarily introduced reference-phase error in the first run. The result of an error-free second run was substracted. In Fig. 4 top sixteen reference-phase steps were used and below four steps. The averaging effect for many steps (as stated above) becomes obvious in this example.

Therefore when only four steps are used it is especially necessary to assure the accuracy of the single phase steps by means of calibration runs. For this purpose the following procedure was programmed and carried out:

One makes four adjustments with the reference-phase values 0,*φ*,3*φ,4φ*. The corresponding intensities may be *I*_{0},*I*_{1},*I*_{3},*I*_{4}, where the index gives the multiplicator of the reference-phase value *φ*. This value can be chosen in such a manner that 4*φ* is just below the maximum elongation of the PZT mirror. Provided that *φ* is approximately known, the four equations,

*φ*:

As will be shown next the averaging of different runs as usual to improve the accuracy can be combined with an error-compensation method in the following way. One makes two runs where the second run has a reference-phase offset of *π*/2. From Eqs. (4) and (5) it follows after some calculation that

*φ*is substituted by [

_{r}*φ*+ (

_{r}*π*/2)] but with a change of the sign of the Fourier coefficients

*C*and

*S*.

If we define

*N*

_{1,2}and

*D*

_{1,2}are the numerators and denominators of the two runs, an improved phase $\stackrel{\sim}{\mathrm{\Phi}}$ can be obtained [this should be expected since

*C′*and

*S′*are smaller than

*C/R,S/R*in Eq. (11) even when the

*ɛ*are distributed randomly]:

_{r}*C′*=

*S′*= 0, i.e.,

*x,y*)-independent phase shift.

This case is of special interest. To show this we assume *ɛ _{r}* to be a power function of the step index

*r*:

*C′*and

*S′*are

*ɛ*contains only terms up to the first order in

_{r}*r*. In this case

*C′*and

*S′*are identically zero because of

In this connection it is interesting to note that the spherical Fizeau interferometer generates an increasing reference-phase error proportional to *r* caused by the axial translation of one of the spherical surfaces (for this purpose see [Ref. 2]). The coefficient is a function of the aperture angle of the translating sphere. With the just discussed procedure the influence of *ɛ _{r}* can be canceled. The application of this procedure to the spherical Fizeau should make corrections unnecessary.

So far we omitted a random error function. The superposition due to Eq. (12) leads to a decrease of the standard deviation by a factor of √2. Therefore, it seems reasonable to apply this averaging method instead of the simple averaging technique used so far because it maintains the ability to reduce the statistical fluctuations and eliminates calibration errors of the reference phase shifter, linear drifts, mechanical relaxation effects, etc., fairly well.

There is the possibility of averaging the results of two runs where the second run is carried out with the *π*/2 reference-phase offset. Also in this case a smoothing effect occurs under the above assumptions. From Eq. (7) one can derive the approximation:

*R*

^{−2}(

*R*is the number of phase steps) and therefore at least 1 order of magnitude below the original values ΔΦ of a single run. The frequency of the remaining oscillations has doubled compared with the oscillations of ΔΦ.

Figure 5 shows the cancellation of the errors of two runs according to the two possibilities discussed. The reference-phase error introduced was chosen as in Fig. 3. Figure 5(a) gives the two runs superimposed on each other with three fringes/diameter. Figure 5 (b) gives the result of an *a posteriori* averaging operation on Φ. In Fig. 5(c) the averaging has been applied to numerator and denominator before the calculation of Φ for a different phase profile with a surface deviation of λ/256 per scale value. Note that the sensitivity has doubled in comparison with Figs. 5(a) and (b).

## III. Errors Caused by Extraneous Interference Patterns

The influence of extraneous light amplitudes on the accuracy has been discussed to some extent by Bruning *et al.*[5] Here, we shall show that there is a Φ dependence of this phase error and propose an elimination technique working also for strong disturbances.

In the presence of extraneous coherent light the intensity distribution is no longer a two-beam distribution. Let three complex amplitudes,

exp(iΦ): | wave field from test arm, |

exp(iφ):_{r} | wave field from reference arm, |

q exp(iη): | wave field from extraneous light, |

interfere with each other; then a three-beam interference pattern results:

*et al.*for Φ = 0 but shows a Φ dependence. This dependence opens up methods for the reduction or elimination of phase distortions caused by extraneous light in the interferometer. For this purpose an additional phase shifter in the object arm is needed. If the phase is shifted by

*π*, an averaging of the Φ data of two runs (one taken with Φ and one with Φ +

*π*) leads to a strong decrease in the amplitude of the disturbing oscillations.

The remaining distortion *δ*Φ is given here without proof:

*q*/2 (

*q*≪ 1) where

*q*is the ratio of the extraneous light to the test arm light amplitude.

The efficacy of the algorithm is demonstrated in Fig. 6. In this experiment a glass plate has been introduced into the interferometer to get a systematic error of known origin. Figure 6(a) shows a single run with extraneous light disturbances; Fig. 6(b) shows the average of two runs where the phase in the object arm has been increased by *π*. This improvement is unfortunately sensitive to the accuracy of the phase offset *π* in the object arm. This can be demonstrated by averaging two runs where the second run is made with a phase offset that is position dependent. This can be arranged by tilting the object surface before the data sampling of the second run. The average is then amplitude-modulated as can be seen from Fig. 7.

Therefore, we favor a total elimination of such distortions by the following procedure:

Take two runs with the phases Φ and Φ + χ, respectively, and calculate separately the numerators and denominators for each run. These quantities are substituted into the following equation:

*nπ*is forbidden since it leads to expressions of the form 0/0. Because of the elimination formalism for axial and lateral defocusing, χ may be position dependent and slightly defocused as long as the above-mentioned condition (21) is not violated. The best results should be expected for χ =

*π*; thus only small deviations should be present.

To verify our statement we again introduced a glass plate into the reference arm, to get strong extraneous fringes, and we made several experiments.

Figure 8 shows a distorted interferogram. The corresponding phase distribution is given in Fig. 9. In Fig. 10 the runs are superimposed one on the other to show the phase shift introduced in the object arm. A total cancellation occurs if the algorithm of Eq. (19) is programmed and executed as shown in Fig. 11. Because the phase shift deviates somewhat from *π* the usual averaging of the two runs does not work well as can be seen in Fig. 12.

The influence of the amount of phase shift introduced can be studied in the following manner. We make two runs where the second run is made with a position-dependent phase shift in the object arm. Here the algorithm of Eq. (19) fails to work in the neighborhood of phase shifts of 2*nπ* (*n* is a whole number). This is shown in Fig. 13.

Figure 14 shows the (*η* − Φ) dependence of the phase error ΔΦ. Although *η* is constant along the section through the interferogram, periodic deviations due to Eq. (16) occur.

This can be seen more clearly in Fig. 15 where a series of pictures with different relative adjustments of the phase to be measured Φ and the phase *η* of the disturbance are given. The CCD array lies in the vertical direction in the interference photograph on the right of Fig. 15 so that Φ is constant or sgn*η* = sgnΦ and sgn*η* ≠ sgnΦ in the order from top to bottom. The deviation of the wave front has been eliminated by subtracting two runs where the second run had a Φ offset of nearly *π*.

## IV. Coherent Noise

Because highly coherent laser light is used some high frequency spatial phase modulation is generated, the so-called coherent noise. Connected with this phase modulation a certain degree of uncertainty occurs because small alterations in the interferometer adjustment may introduce different phase modulations. High frequency phase modulation becomes disturbing when the two interfering waves take different ray paths through the whole setup.

Within the interferometer (beginning and ending at the beam splitter) the two waves have different ray paths resulting in high frequency Φ variations even in the case of parallel adjustment (see Fig. 1).

In addition there are differences due to relative tilts or slopes of the interfering wave fronts while passing the optics preceding and following the interferometer. Small glass inhomogeneities or dust particles on the surfaces of optical elements lead to diffracted waves which interfere with the background wave. Although the diffracted wave is weak, the wave front resulting from the interference of the scattered wave field and the background is no longer smooth. When the slope of two wave fronts differs, the diffracted waves are sheared relative to each other in the observation plane. The amount of shear is proportional to the slope difference of the two interfering wave fronts at some point in the interferometer exit.

To get an insight into the problem at stake we make a few estimations:

Each phase distortion results in intensity modulation (referred to in the literature as dust diffraction[6]). So one can derive the modulation amplitude of the phase from the intensity fluctuation of one of the beams. The complex light amplitude *u _{b}* of the background wave (Fig. 16) is superimposed by the amplitude of the scattered wave

*u*having any phase

_{n}*φ*within the interval 0…2

_{n}*π*relative to the background wave.

The resulting complex wave amplitude *u _{r}* has a phase within the interval Φ ± ΔΦ. The maximum phase modulation caused by diffraction at disturbances is ΔΦ = |

*u*|. Because of the weakness of the scattered field we neglect the interference between the waves from different scatterers and restrict the discussion to the case of two-beam interference between the background and one scattered wave. So one gets

_{n}/u_{b}*φ*is the phase of the scattered wave and

_{n}*I*are the background and scattered intensities, respectively. The maximum deviation of

_{b},I_{n}*I*from the mean is

*I*

_{max}−

*I*

_{min}= 4|

*u*| |

_{b}*u*|. In this approximation the mean intensity is

_{n}*I*, so the following relation holds:

_{b}The detected spectrum of spatial frequencies of the disturbances is owing to the number of independent samples taken by the detector lying in the region of 1/*P* < *ν* < 1/*p*, where *p* is the period and *P* is the length of the CCD line array.

From Fig. 17 and from an estimate of the geometry one may conclude that there is a dominant spatial frequency (here in the range of 10/*P* – 15/*P*) of the phase modulation in the interfering wave fronts. For the phase of a distorted plane wave we assume here a periodic oscillation of Φ:

*x*one to another in the plane of observation. The phase shift 5Φ occurring as distortion of the measured phase data due to the shear is

*s*of the plane of interference from the scattering obstacle or, more generally, from the image of the obstacle. For the sake of simplicity the slope difference

*α*may be introduced by a tilt of one arm relative to the other arm of the interferometer. In this case

*α*= (

*N*λ)/2

*P*, where

*N*is the number of fringes in the field,

*P*is the length of the array, and λ is the wavelength. The peak-to-valley deviation

*δ*Φ

*then becomes*

_{pυ}*s*< 200 mm. With 10/

*P*<

*ν*< 15/

_{m}*P*and

*N*= 10 fringes per array length

*δ*Φ

*lies in the region 3.6 ΔΦ <*

_{pυ}*δ*Φ

*< 10 ΔΦ. From Fig. 19 one gets*

_{pυ}*δ*Φ

*= 0.5 rad. From Fig. 17 we have ΔΦ = 0.08 rad, so there is an approximate correspondence between these values and the estimates given above.*

_{pυ}Hitherto, only the contributions from disturbances generated outside the interferometer were taken into account. From the scatterers within the interferometer, especially from the optics in the test arm, nearly the same contribution to the phase modulation originates. This can be studied if we assume a parallel adjustment (shear Δ*x* = 0) of the interfering beams. In this case the expression for the intensity distribution can be factorized so that the influence of the scatterers outside the interferometer on the phase is canceled by the algorithm of Eq. (1). The remaining disturbances are given in Fig. 20 having approximately half of the amplitude in comparison with Fig. 19.

The considerations so far hold in the same way for conventional fringe evaluation interferometry when laser light is used. It also is handicapped by coherent noise from the imaging optics.[8]–[11]

Because coherent noise disturbs the measurement in any case it, therefore, seems advisable to suppress high frequency phase variations of the wave front which are caused by dust diffraction. This can be achieved partly by filtering and partly by averaging the light intensity in lateral dimensions.

Filter operations can be carried out in the focal plane at the interferometer output. Because in the last consequence only 32 × 32 pixels are used for a 2-D evaluation, the stop may have an angular dimension of 32 λ/field diameter. This restriction of the plane wave spectrum leads to a smoothing effect as shown in Fig. 21.

For the purpose of averaging in lateral dimensions an intermediate image of the surface to be tested is picked up by a moving scatterer. This is imaged in a further step on the detector (CCD or other). In this incoherent imaging process it is easy to curtail the MTF of the whole process by defocusing. In Fig. 22 the result with and without such defocusing is shown.

Several of the measures mentioned so far should be combined. This is especially important if one is urged to use high coherent light from a laser.

To summarize, these measures are (1) measuring in parallel adjustment; (2) using a stop in the exit pupil of the interferometer; (3) averaging by a moving scatterer which is out of focus in an intermediate image plane following the interferometer exit (in some cases the special detector regime provides a certain amount of lateral averaging[10]); and (4) using well-corrected beam shaping optics in the test arm of the interferometer.

Furthermore, the optics in the whole imaging system should be clean and free from scratches and inhomogeneities to the highest possible degree.

## Conclusions

Three types of errors have been considered.

Errors caused by inaccurate reference phases (caused by mirror movements or by thermal drifts, etc.) can be diminished by (a) use of a parallel adjustment and only small wave aberrations in the interferometer; (b) use of the averaging technique contained in Eq. (12); (c) calibration of the reference mirror driver; and (d) use of multiple sampling (*R* > 4).

Furthermore, the averaging technique contained in Eq. (12) can be used to eliminate errors inevitably introduced by shifting the spherical test glass in a digital spherical Fizeau interferometer because the reference phase depends linearly on the movement[2] [as required by Eq.(14c)].

Errors introduced by extraneous fringes can be totally eliminated by making two runs, where the second has an object phase offset of *π*, and by a suitable combination [due to Eq. (19)] of the measured intensity values.

Errors generated by dust diffraction can be controlled by measurements in parallel adjustment and by filtering and lateral averaging of the intensity data.

With our present interferometric setup we could attain a repeatability of λ/200 in the ±3*σ* range, respectively, of λ/600 in the ±*σ* range as can be deduced from Fig. 23.

## Appendix

Concerning Eqs. (11)–(12a) we give here a derivation of Eq. (12a) on the basis of Eq. (12) and the described experimental procedure.

Let the intensities during the first run be denoted by ${I}_{r}^{\phantom{\rule{0.1em}{0ex}}(1)}$ and during the second run by ${I}_{r}^{\phantom{\rule{0.1em}{0ex}}(2)}$. The actual reference phases given by the translation of the reference mirror are indicated by ${\psi}_{r}^{\phantom{\rule{0.1em}{0ex}}(1)}$ and ${\psi}_{r}^{\phantom{\rule{0.1em}{0ex}}(2)}$ respectively [see Eq. (2)], where according to our assumptions the following holds:

*φ*are the ideal reference phase values and ${\varepsilon}_{r}^{\phantom{\rule{0.1em}{0ex}}(1)}$, ${\varepsilon}_{r}^{\phantom{\rule{0.1em}{0ex}}(2)}$ are the errors of the reference phase values for the two consecutive runs.

_{r}Due to Eqs. (12) and (1) we have

*x*and cos

*x*with the approximations cos

*ɛ*= 1; sin

_{r}*ɛ*,. =

_{r}*ɛ*:

_{r}As pointed out in Eqs. (13)–(14c) reference-phase errors
${\varepsilon}_{r}^{\phantom{\rule{0.2em}{0ex}}(1)}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.1em}{0ex}}{\zeta}_{1}\phantom{\rule{0.2em}{0ex}}(r\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}1)$ and
${\varepsilon}_{r}^{\phantom{\rule{0.2em}{0ex}}(2)}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.1em}{0ex}}{\zeta}_{1}\phantom{\rule{0.2em}{0ex}}(r\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}1\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}a)$ (*r* is a whole number) lead only to an additional offset term arctan*ɛ′*. For arbitrary *R*(*R* > 3) the cumulative error shall be denoted as *ɛ _{R}*. Then the slope

*ζ*

_{1}of the error function is

*a*=

*R*/4 [because (2

*π*)/

*R*= (

*π*/2)/

*a*], if the reference phase is driven through one period of the interference pattern, the offset becomes

*f*(

*x,y*) =

*a*+

*bx*+

*cy*+

*d*(

*x*

^{2}+

*y*

^{2}), with

*a, b, c*, and

*d*as constants. This ambiguity is eliminated by a least-squares algorithm. Therefore, the offset of the type of Eq. (A7) is eliminated (remaining systematic error < λ/300) even if

*ɛ*depends on (

_{R}*x*

^{2}+

*y*

^{2}) as is the case with the spherical Fizeau interferometer

^{2}(arctan

*ɛ*is approximated by

_{R}*ɛ*), at least as long as the numerical aperture of the surface is below 0.6.

_{R}A time-saving algorithm with *R* = 4 and only 5 intensity values *I*_{1}…*I*_{5} corresponding to reference phase values *φ _{r}* = 0,

*π*/2,

*π*,(3/2

*π*),2

*π*can be derived from Eq. (A4), i.e.,

*ɛ*

_{4}/2). Because only simple arithmetic is used this formula is especially suitable for

*μP*use.

## Figures

## References

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