Interferometric null test of a deep parabolic reflector generating a Hertzian dipole field

Interferometric null test of a deep parabolicreflector generating a Hertzian dipole field

Gerd Leuchs, Klaus Mantel,* Andreas Berger, Hildegard Konermann,Markus Sondermann, Ulf Peschel, Norbert Lindlein,

and Johannes SchwiderFriedrich Alexander University of Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany

*Corresponding author: [email protected]‑erlangen.de

Received 3 July 2008; revised 9 September 2008; accepted 12 September 2008;posted 15 September 2008 (Doc. ID 98280); published 13 October 2008

We report on interferometric characterization of a deep parabolic mirror with a depth of more than fivetimes its focal length. The interferometer is of Fizeau type; its core consists of themirror itself, a sphericalnull element, and a reference flat. Because of the extreme solid angle produced by the paraboloid, thealignment of the setup appears to be very critical and needs auxiliary systems for control. Aberrationscaused by misalignments are removed via fitting of suitable functionals provided by means of ray tracingsimulations. It turns out that the usual misalignment approximations fail under these extreme condi-tions. © 2008 Optical Society of America

OCIS codes: 260.0260, 120.0120, 120.3180, 220.4840.

1. Introduction

In previous publications [1,2] an experiment was pro-posed which shall give insight into the absorption ofsingle photons by single atoms exposed to the fielddistribution of a linear dipole. Since the latter irradi-ates into almost the full solid angle, the atom has tobe excited from full space in the ideal case. In [1,2] itwas proposed that this ideal case is closely approxi-mated by an atom that is located in the focus of adeep parabolic mirror that is illuminated by a planeradially-polarized wave with a specially designed ir-radiance. Compared to other approaches [3], the fo-cused wave field is then a good approximation to therequired Hertzian dipole field.However, parabolic mirrors exhibit aberrations in

practice, resulting in a nonuniform phase distribu-tion after reflection off the mirror. This leads to de-formations of the ideal focal field distribution and,thus, to a reduced absorption probability. Therefore,the aberrations of the parabolic mirror have to bemeasured and corrected.

Here we report the realization of such a measure-ment by means of interferometry. Since parabolicmirrors are also discussed for microscopy applica-tions [4], we believe that the reported results are alsoof practical relevance for other experiments invol-ving parabolic mirrors.

Interferometric surface tests of stigmatic asphericslike a parabolic mirror can be carried out in a nulltest configuration. In the case treated here, the nullelement is a sphere, since the parabolic reflectortransforms a plane wave into a spherical one. Thisresults in a double pass geometry for the null test,which is realized here with the help of a Fizeau in-terferometer. This interferometer ensures invarianceagainst polarization distortions under the assump-tion that radially-polarized light is used. This stateof polarization has to be taken to avoid phase varia-tions which are caused by the polarization depen-dence of the Fresnel reflection of metals, since theparabolic mirror used here is made of aluminum.The last constraint requires a testing setup that isfree from polarizing elements, such as quarter-waveplates or polarizing beam splitters. Hence, rotationalsymmetry has to be maintained in the test geometry.

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Tolerance calculations have shown that thesuccess of the experiment depends strongly on theremaining wave aberrations produced by the para-bolic reflector at the wavelength of operation. Thewave aberrations of the generated dipole waveshould be below λe=4 P/V, which is well-known fromscalar calculations but proves to be valid for veryhigh aperture angles as well. This requirementhas two aspects: (1) the surface deviations fromthe ideal parabolic have to be smaller than λe=8cosðθðρÞÞ, where θðρÞ is the incidence angle on theparabolic surface at incidence height ρ, and (2) theincoming plane radially-polarized wave has to bealigned on axis with utmost accuracy.Since the wavelength λe to be used in the photon

absorption experiment is in the 250nm range, thesurface deviations of the mirror have to be knownwith a high absolute accuracy.

2. Interferometric Test Setup

Taking the symmetry requirements into considera-tion the setup shown in Fig. 1 should fulfil the neces-sary conditions to obtain a polarization insensitivenull test.The paraboloid was produced by diamond turning

of an aluminum work-piece (Fraunhofer IOF, Jena).Because of the manufacturing method, the mirrorwill show a typical microstructure or waviness withan rms figure of about 25nm, which is good enough toallow for interferometric tests and to use in theexperiment without further polishing. Therefore, apolarized He–Ne laser at λt ¼ 633nm is used. Theparaboloid has a focal length of 2:1mm and an aper-ture diameter of 20mm so that the resulting halfaperture angle is nearly 135°, i.e., much larger than90°. This corresponds to a depth of 5.6 times the focallength.The on-axis-alignment requirement mentioned be-

fore results from the strong violation of the sine con-dition through a parabolic mirror [5]. To support thealignment process, a plane reference surface isturned in the rim region of the parabolic mirror inthe same state of adjustment on the turning lathe.In addition, the wavelength of operation (λe ¼252nm), where the dipole transition occurs andwhich defines the required quality of the surface,is much smaller than the wavelength used for the in-terferometric testing (λt ¼ 633nm). Hence the re-quirements on the test accuracy are even morestriking.

Because of the prevailing symmetry the normal ofthe plane reference surface will be coincident withthe optical axis which is also the axis of symmetryof the parabolic mirror within the accuracy of thelathe.

For the illumination of the core Fizeau interferom-eter a polarized He–Ne laser is used. The Fizeau in-terferometer consists of a plane reference mirror, theparaboloid, and a steel sphere having a diameter of1mm. For the adaptation of the mean intensity of thelaser beam to the dynamic range of the detector, arotatable half-wave plate (HWP) is used in combina-tion with a polarizer in front of the beam expandingtelescope. The emerging plane and linearly polarizedwave passes the LCD-polarization transformer(Pola-trafo) [6] and, afterwards, a spatial filteringtelescope in focal plane F1 of which a pinhole is usedto remove distortions from the beam. The emergingradially-polarized wave traverses a plane plate ser-ving as a beam splitter and enters the beam expand-ing telescope in front of the Fizeau interferometer(the tilt of the beam splitting plate in Fig. 1 is largelyexaggerated). One side of the interferometer isformed by the flat back plane of a glass plate. Thefront surface of this glass plate has a small wedgeangle to separate reflections from the back surfacewhile still not disturbing the polarization state ofthe light. In the case of a correct adjustment of thesteel sphere and the parabolic mirror, the light re-flected at the surface of the steel ball retraces itsway back via the paraboloid and interferes withthe wave reflected from the back side of the glasswedge. Emerging interference fringes are observedvia the beam splitter by an imaging telescope anda CCD array. Furthermore, the spatial filter F3 inthe common focal plane of the imaging telescope en-ables a suppression of parasitic light generated bythe reflection at lenses and at the back surface ofthe beam splitter. The reference flat is mounted onthree calibrated piezo stacks that enable axial shiftsand, thus, allow for phase shifting evaluations of theinterference pattern.

As a spherical reference surface, a steel ball isused, which is glued to the conical tip of a 1mm steelrod (sphere and rod provided by Carl Zeiss, Göttin-gen). The conical tip has a diameter of about300 μm and allows for measurement within the fullaperture of the paraboloid. The paraboloid has a cen-tral hole on axis with a diameter of 1:5mm. This holecan be used to adjust the reference ball from thebackside (see Figure 2). For the positioning of the ballsurface a 3D translation stage with piezo transdu-cers is implemented allowing for a touchless fine tun-ing of the sphere. In the case of a correct adjustment,the center of curvature of the sphere coincides withthe focus of the paraboloid.

3. Aligning Method for the On axis Position of theParaboloid

It is well known [5] that a parabolic mirror violatesthe sine condition or, in other words, it shows strongFig. 1. Fizeau interferometer for testing a parabolic mirror.

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comatic aberrations for off-axis points in the focal re-gion. Therefore, it is necessary to align the parabolicmirror very accurately on axis. This is difficult sincethe deep parabolic mirror has an extremely highaperture angle in order to get as close as possibleto a solid angle of 4π for the wave converging tothe focus. In our case the solid angle covered bythe spherical wave is 3:4π. This corresponds to94% coverage of the weighted solid angle for aHertzian dipole oscillating on the optical axis ofthe mirror [1].Here it is substantially helpful that the parabolic

surface and the plane surface is machined with thesame alignment on the diamond turning lathe. Be-cause of the symmetry it is guaranteed that the nor-mal of the reference flat in the outer rim region is onaxis with the axis of symmetry of the parabolic sur-face. If a plane wave is reflected at this reference flatit can be superimposed with another plane wave, thenormal of which coincides with the light direction ofthe illuminating laser. An optical device that pro-duces such a plane reference wave representingthe direction of the impinging laser beam is a cat’seye mirror, which can be realized by an achromatwith a plane mirror positioned in its focal plane. Alight ray entering this cat’s eye mirror will be re-flected parallel to itself, realizing in this way a refer-ence for the direction of the incoming laser light.As is shown in Fig. 1, the beam splitter plate pro-

duces an additional arm of a Michelson interferom-eter beside the so far discussed Fizeau for the test ofthe parabolic mirror. The cat’s eye mirror is placed inthis additional reference arm. After reflection, the re-flected light from the cat’s eye mirror enters the ima-ging telescope and represents the reference directionfor the adjustment of the parabolic reflector. The cat’seye wave interferes with the light reflected at the flatsurface in the outer rim region of the parabolic mir-ror. The correct alignment is achieved if the twobeams in the Michelson are in coincidence in thewhole space, i.e., at finite distance, here at the front

surface of the telescope, and at infinite distance, herein the focal plane of the first telescope lens. Coinci-dence at finite distances can be achieved through lat-eral translations of the cat’s eye reflection setup andcoincidence at infinity by turning the paraboloid to-gether with the ball sphere around two orthogonalaxes. If this is achieved, then fluffed out fringes forthe interference pattern produced by the cat’s eye re-flection and the reflection from the rim flat of theparabolic mirror will occur (see Fig. 3, left).

The sensitivity of this interferometric criterion al-lows for an accuracy of a few arc seconds, which issufficient to be free from comatic contributions inthe final test of the parabolic surface. The latterhas been confirmed by computer simulations withthe help of our program RAYTRACE (Optics DesignProgram RAYTRACE, www.optik.uni‑erlangen.de/odem) (see Fig. 3, right).

4. Removal of Adjustment Aberrations from the TestResult

In interferometric tests the resulting interferencepattern is always biased by aberrations caused bythe misalignment of the component under test. Inthe most simple case of planeness testing, the align-ment aberrations are linear functions of the space co-ordinates. With increasing complexity, quadraticand, in general, higher-order terms have to be takeninto account. Even in the case of sphericity tests, theapproximation following from the displacement the-orem [7] holds only for small surface apertures. Thiscan be demonstrated by simulating the misalign-ment aberrations for certain shifts of the referencesteel ball. Figure 4 shows the expected interferencefringes for axial (left image) and lateral (middle im-age) shifts by 1 μm.

Furthermore, the planeness of the illuminatingwavefront has to be controlled as well. As asimulation has shown (see Fig. 4, right) a radius ofcurvature of 100m gives rise to a regular aberration

Fig. 3. Superposition of the plane waves coming from the cat’s eyereflector with the nearly plane wave reflected from the outer rim ofthe paraboloid. The aberrations in the central part are due to in-terference of the test wave with the cat’s eye reference. The re-maining aberrations are due to an imperfect cat’s eye reflector.But a simulation of the wave aberrations (on the right) of anoff-axis angle of 1arcmin for the illuminating beam shows thatthe achievable accuracy of this adjustment is satisfactory (P/V:0.02 wavelengths) since, in the left hand fringe pattern, only a mis-alignment of a few arc seconds is indicated.

Fig. 2. Core of the Fizeau test setup: Comprised of the parabo-loid, the plane reference surface, and the spherical steel sphereas a null element for the spherical wave formed in the focal planeof the parabolic mirror.


in the order of 10−3 waves (after subtracting defocus).With the help of shearing tests it is possible to carryout the focus adjustment with sufficient reliability.Figure 5 shows typical examples of the misalign-

ment patterns which occur even within the para-meter tuning range (about 15 μm) of the piezotransducers for the x, y, and z coordinates of the steelball reference. This indicates the extreme sensitivityof the test setup against very small misalignmentsand illuminates the difficulties of the experiment.For aspherics the alignment functionals are usual-

ly derived from the projection of the shift vector ontothe surface normal. For the general case of nonnor-mal incidence, the following equation holds [8] forthe misalignment induced aberrations

Φmis : Φmis ¼ −4kðd · nÞðe · nÞ; ð1Þ

where d is the misalignment vector, n is the normalvector field of the surface under test, e is the unit vec-tor field of the incident light rays, and k is the wave-number.The simulation reveals that an analysis based on

the approximation of Eq. (1) will not provide the re-quired accuracy in the extreme case at hand. Thiscan be inferred from the simulation of a 1 μm axialmovement of the paraboloid from the ideal position.There remains a numerical residuum of 0:07wavesP/V which is not tolerable (see Fig. 6).Therefore, an approach relying on the method of

virtual motions has been exploited where the aberra-tions are calculated using RAYTRACE and a Zernikefit of degree 12 to obtain suitable functionals Ziðx; yÞ.There are five adjustment parameters for the para-

bolic mirror, i.e., three translations δξ, δη, and δζalong the three coordinate axes, two rotations Ωxand Ωy, and a tilt contribution Lðx; yÞ from the planereference surface:

Φmis ¼ Zξðx; yÞδξþ Zηðx; yÞδηþ Zζðx; yÞδζþ ZΩx

ðx; yÞΩx þ ZΩyðx; yÞΩy þ Lðx; yÞ;Lðx; yÞ

¼ aþ bxþ cy: ð2Þ

The free parameters are chosen such that the de-viations between the phase calculated on the basis ofEq. (2) and the experimentally determined valuesare minimized.

Since the sensitivity of the aberration balance isfor all three translations of the paraboloid—due tothe very large surface aperture of the surface undertest—it is straightforward to select the virtual shiftsin the simulation to be equal. The validity of this ap-proach is shown in Fig. 7 for the same axial transla-tion as in Fig. 6.

5. Measuring Results

After correct alignment, the measurement startswith the selection of a data mask for the evaluationof the aberration data of the paraboloid. This maskselects an annular field for the valid data (see Fig. 8,left). There is a certain degree of arbitrariness in thisselection process, which is discussed in detail lateron. After the field has been selected, the phaseshifting evaluation is done and results in thewrapped phase of the aberrations (see Fig. 8, right).After the unwrapping process, the misalignment

Fig. 5. Different experimental adjustment states of the steel sphere relative to the parabolic mirror: On the left, two z adjustments; on theright, lateral displacements in x and in x, y direction.

Fig. 4. Simulated interferograms due to alignment aberrations obtained by the programRAYTRACE: On the left, axial shift of 1 μmof theparaboloid (or steel sphere) against the interferometer frame; in the middle, lateral shift of 1 μm of the paraboloid (or steel sphere); on theright, collimation error (R ¼ 100m) of the impinging wavefront.


contributions are removed by the ray-tracing sup-ported-fitting process we discussed in Section 4 (see Fig. 7).Furthermore, the phase aberration is divided by afactor of 2 to compensate for the double pass geome-try of the measuring setup. Figure 9 shows theresulting phase aberrations introduced by theparabolic mirror in a single pass. Since the aberra-tion data also contain irrelevant phase disturbancesdue to dust diffraction patterns and spurious fringes,it is very helpful to use a Zernike fit for the clear dis-play of the most relevant aberrations of the parabo-loid (see Fig. 10).Still, these values contain a contribution from an

angle-of-incidence dependent phase shift at the me-tallic mirror surface. At the measurement wave-length of 633nm, however, this contribution issmaller than 0:06λt and is, therefore, neglected.Furthermore, it should be kept in mind that the test

configuration mimics the final experiment very clo-sely. So this minor aberration adds only to the totalof aberrations which have to be corrected by the useof a phase plate.

6. Three-Beam Interference

During our first experimental tests we noticed thatdue to the high reflectivities of the metallic surfacesa third wave is generated by double reflection at theparabolic mirror and the steel ball in the Fizeaucavity. This wave disturbs the two-beam characterof the interference fringes and needs a more detaileddiscussion.

To introduce the problem let us discuss a simpleFizeau interferometer for planeness testing withtwo plane plates. One of these plates has a highly re-flective coating on one surface and the other plateshall be uncoated. The latter surface shall be usedas the reference surface (see Fig. 11). The reflectivityof an uncoated glass surface is of the order of 4%.

In this arrangement three strong waves are pro-duced resulting in three-beam interference fringes.As can be deduced from Fig. 11, a small tilt of thereference surface will produce two waves with an in-tensity of about 4% (designated in Fig. 11 with reflec-tivities R1 and R2) besides the strong reflex from thecoated surface (reflectivity Rh ≈ 1). One is resultingfrom the direct reflection at the uncoated referencesurface. The other is first reflected at the highly re-flective surface, then at the reference surface, andback-reflected from the reflective-coated surfaceagain. The latter wave is a parasitic wave which

Fig. 8. On the left: interference fringes shown together with theselection mask which has to be centered and the outer and innerdiameter matched to the dimension of the valid data field. On theright: wrapped data giving an indication of the quality of the data.

Fig. 9. Relevant phase aberration data introduced by the para-boloid in the null test configuration of Fig. 1: On the left, grayscaleplot of the deviations; on the right, pseudo 3D-plot of the deviations(all values are in waves).

Fig. 6. Simulated misalignment in axial direction: on the left,1 μm axial shift of the paraboloid; on the right, residuum due tothe approximation of Eq. (1) (all values are in waves).

Fig. 7. Simulation result showing the remaining residuum afterremoval of an axial shift of 1 μm, as in Fig. 6 before, but using themethod of virtual shifts due to Eq. (2) and a Zernike fit of the de-gree 12 to the aberrations obtained by ray tracing (all values are inwaves). Now, the P/V is only 0.0039 waves.


has the opposite tilt to the wanted reference wave-front. The problem for phase shifting evaluationsarises from this opposite tilt of the two weak wavessince an axial movement of the reference flat causesopposite phase shifts for these waves which maycause a nonmodulating intensity pattern obstructingthe calculation of the phase to be measured.In fact, such a Fizeau interferometer is formed by

the reference surface and the diamond turned planesurface in the rim region of the paraboloid for adjust-ment purposes. Fortunately, the interference patternin this region is only used for the adjustment and notfor the test of the parabolic surface itself.A slightly different situation occurs if one consid-

ers the parabolic surface in combination with thesteel ball reference surface because then allsurfaces beside the reference flat are rather highlyreflective. Here, three reflections from highly-reflective surfaces, two from the aluminum parabolicsurface, and one from the steel ball surface are con-tributing to the intensity of the test beam. For thefollowing discussion only the ratio of the complex am-plitudes is relevant, which allows for setting the in-coming intensity to 1. The three waves generated inthe setup are: (1) the reference intensity that is pro-portional to Rgl, (2) the test-beam intensity that isproportional to ð1 − RglÞ2R2

AlRSt, and (3) the parasiticintensity that is proportional to ð1 − RglÞ2RglR4

AlR2St, where Rgl is the reflectivity of glass, RAl is

the reflectivity of aluminum and RSt is the reflectiv-

ity of steel. The three reflections produce three sepa-rate spots in the common focal plane of the imagingtelescope if a tilt of the reference plate has been in-troduced (see Fig. 12). Because of the high number ofreflections of the parasitic beam, its intensity isstrongly reduced.

If the steel ball reference sphere was substitutedby a sphere made from an absorbing glass, the para-sitic term would have been negligible and true two-beam evaluations would have become possible.Unfortunately, so far it has not been possible to finda supplier for a high-quality ball sphere made fromabsorbing glass with a diameter of about 1mm. Con-sequently, the influence of such a parasitic wave hasto be investigated.

There are different possibilities to suppress theparasitic wave: (1) either through the placement ofan absorber or partially reflective surface in the in-terferometer cavity, which can also help to improvethe contrast of the fringes used for the measurement,or (2) through a relatively large tilt of the plane re-ference Fizeau surface in combination with a spatialfiltering operation in the imaging branch of the inter-ferometer.

In the case of the first approach there is the pro-blem of additional aberrations caused by the absor-ber. This can be inferred from a measurement withan inserted neutral glass filter in the cavity(see Fig. 13).

The improvement of the contrast together with thereduction of parasitic amplitude is obvious but thestrong aberrations of the NG-filter screen the aberra-tions of the paraboloid. We did not follow this ap-proach, because the quality of the available filterswas not high enough.

The second approach uses a strong tilt of the refer-ence mirror, which separates the waves in the spatial

Fig. 10. Zernike fit of the relevant phase aberration data: On theleft, contour line plot; on the right, pseudo 3D-plot (all values are inwaves).

Fig. 11. Demonstration example for the occurrence of three-beaminterference (waves proportional to R1 ¼ Rgl;ð1 − RglÞ2Rh, and R2)in a plane surface Fizeau interferometer with a highly reflectivetest sample (Rgl is the reflectivity of glass, Rh is the reflectivityof the coated surface). In addition, an even weaker wave R3 is gen-erated that can be ignored in this context. Reflections from theback surface of the reference plate are not shown.

Fig. 12. Intensities detected in the focal plane of the imaging tele-scope. The central spot stems from the parabolic mirror in the nulltest arrangement. The spot in the upper half of the image belongsto the reference wave and the spot below to the parasitic intensityof the doubly-reflected beam at the null test configuration (para-sitic beam). The images on the left and in the middle differ by theamount of tilt of the reference plate. The situation in the middle ischosen to be able to block the parasitic wave that is realized on theright. The parasitic intensity has been suppressed by an opaquescreen (on the right).


frequency space (see Fig. 12, right). Such a separa-tion is possible in the common focal plane of theimaging telescope. There, a spatial filter can bepositioned which transmits only the reference andthe test wave and screens the parasitic wave.Furthermore, it offers also a balancing of the inter-fering beams through an amplitude filter. Since thisapproach leaves the interferometer cavity as it is, theinterference pattern is not impaired by additionalaberrations as long as the optics of the imagingtelescope fulfils the sine condition and the—possible—amplitude filter is free from aberrationsin the microregion. Both conditions are met in oursetup. However, a sufficient tilt might cause addi-tional aberrations in the collimating and imagingoptics because now the light paths of the two inter-fering waves no longer coincide in the imaging optics,which is commonly not the best solution to guaranteehighest accuracy [9]. Therefore, this approach needsfurther investigation (see below).Just to demonstrate the impact of this parasitic

wave, a phase shifting evaluation of the interferencepattern between the test and the parasitic wave hasbeen performed (see Fig. 14), which can easily beachieved by selecting these two waves in the spatialfrequency domain. It is obvious from the unwrappeddeviation result freed from a linear term that thecontrast of the fringes in the lower part of the aper-

ture was below a threshold maintained in the evalua-tion to suppress invalid pixels.

In Fig. 15 an evaluation of an interferogram freefrom the contribution of the parasitic wave is shown.The reference surface was tilted to separate and tofinally block the parasitic wave in the focal planeof the imaging telescope. Consequently, the filteredimage shows a high spatial frequency. The intensityand raw phase plots in this figure show the dominantlinear aberration term due to the tilt of the referencemirror. Already the P/V values show similarity to theresult given in Fig. 10 which was made without car-rier frequency but with parasitic wave.

Figure 16 shows the difference between the mea-suring results using a tilt of the reference surfaceand a measurement in parallel adjustment whichsuffers from the parasitic wave amplitude. Both re-sults agree quite well within the scope of measure-ment accuracy, so that, contrary to our first guess,the impact of the parasitic wave on the ultimate ac-curacy can be neglected in the face of other error con-tributions. So, in the further discussions, allmeasurements were made in parallel adjustment(minimum tilt fringes).

7. Systematic Error Sources

The accuracy of the test of the paraboloid stronglydepends on the deviations of the ball reference

Fig. 13. Measurement with NG-filter with 10% transparency. It can be inferred from the video picture (upper part) that the NG-filterintroduced strong aberrations which screen the phase aberrations introduced by the parabolic mirror (all values are in waves).


sphere from a truly ideal sphere and on the devia-tions of the plane Fizeau reference surface. In ourinterferometer, a plane surface with λt=100 P/V de-viation has been used. A Fizeau test against a secondsurface of similar quality is shown in Fig. 17, whichsuggests that the flatness deviations are very wellbelow the envisaged accuracy level of λt=50 P/V.It should be mentioned that the remaining surface

deviations also contain a quadratic contribution froma global curvature. This contribution can be ne-glected since such terms are greatly reduced throughthe fitting of regular functionals representingdefocus terms.Furthermore, the phase shift technique provides a

broad margin of sensitivity [9] for the phase mea-surement. This can be demonstrated by performinga repeatability test (see Fig. 18). However, the selec-tion of the active data field is most critical concerningthe repeatability. The data field is selected via inter-active matching of an annular mask with the rim ofthe fringe field. On the one hand, this software pro-cedure offers a unique data field free from fluctua-

tions in the validity of the data but, on the otherhand, this procedure is prone to systematic aberra-tions due to fitting operations necessary for the re-moval of misalignment functionals. This can bedemonstrated if the aberrations of two measure-ments are subtracted after independent adjustmentshave been done (see Fig. 19) and Zernike fits of de-gree 12 have been applied.

Now, let us discuss the case of the ball null, whichis even more involved. Since an absolutely testedsphere does not exist, at least not with the extremeparameters necessary for the test of the paraboloid,it is necessary to undertake efforts to improve the re-liability of the achieved test data.

In a more general sense, the test geometry shownin Fig. 1 can also be used to test the sphere where theparaboloid is just used as a wavefront shaper. Pro-vided the deviations of the plane reference surfacefrom true flatness can be neglected, the measured de-viations have to be attributed to the paraboloid andthe reference sphere, since these two surfaces arethe most precarious ones. The elimination of the

Fig. 14. Evaluation result of test-beam with parasitic beam: upper part, video image; lower left, wrapped phase; lower right, unwrappedphase after removal of linear aberration term. Please note that, in the lower region, the contrast of the resulting interference pattern wasbelow a threshold used in the software to avoid false phase values. In the lower part of the unwrapped deviation picture, a failure of theunwrapping process can be seen which is a direct result of the poor contrast of the parasitic fringes.


deviations of the spherical ball presupposes theknowledge of the absolute deviations of this spherefrom an ideal one. Absolute sphericity test methods[10–16] have been developed for spherical surfaceswith surface apertures D=2R (surface diameter / 2times the radius) below 1.The usual procedure to obtain absolute deviation

data of plane or spherical surfaces relies on relativetests where the test piece or the normal surface un-dergoes certain movements with respect to the rigidinterferometer frame [10]. If differences of measureddata are considered, the aberrations of the resting in-terferometer components are cancelled. But thisalone does not deliver absolute data, i.e., deviationsfrom the ideal surface. In the case of the sphericity

measurements, these data do not comprise the abso-lute values for the radius of curvature which is, in theprevailing context anyway, without relevance.

However, the nearly full coverage of the 4π-solidangle makes a direct transfer of known absolute testsin this case highly involved. As an example for an ab-solute sphericity test, let us discuss the case ofspherical proof glasses. Absolute data or more quali-tatively spoken information on the deviations ofproof glasses from an ideal sphere can be derivedfrom measured data including rotations and lateralshifts of two complementary proof glasses relative toeach other. The translation is carried out as a rota-tion around the center of curvature, which is equiva-lent to a tilt in the plane of the surface, followed by alateral shift without the introduction of unwantedfringes. Schulz [17] has developed a mathematicalform to describe this procedure. But here the extremecurvature will be a severe challenge because the co-ordinate system is no longer Cartesian. A differentapproach to obtain absolute data has been put for-ward by Evans and Kestner [18] and has beenstepped up to an absolute sphericity test by Seitz[19]. In this approach the sphere is rotated aroundthe axis of symmetry and the received aberrationdata are summed up, reducing the nonrotationalsymmetric aberrations to a negligible level. The non-rotational aberrations can then be obtained from the

Fig. 15. Evaluation result of the phase aberrations introduced by the paraboloid with tilted reference surface and suppressed parasiticwave: Above, fringe pattern on the CCD-chip (left) and wrapped phase after phase shifting evaluation (right); below, Zernike fit of the waveaberrations of the test beam (all values are in waves).

Fig. 16. Difference of the Zernike evaluations resulting from twomeasurements—one with and the other without tilt fringes (allvalues are in waves).


difference data of the rotationally averaged and asingle measurement out of the set of data. In thissense the paraboloid serves only as auxiliary opticsfor the measurement of the nonrotational symmetricaberrations of the extreme sphere. The rotationalsymmetric aberrations of the ball sphere can be ob-tained from shear data which can be achievedthrough rotation of the sphere around the center ofcurvature [13]. Unfortunately, such a rotation cannotso easily be implemented in our case. This restrictsthe procedure to a rotational averaging, meaningthat the rotational symmetric aberrations of thespherical ball reference remain in the deviation dataof the paraboloid. The averaging of measured datausing different spheric lenses could be exploited asan alternative approach for the improvement of

the reliability of the aberration data under the as-sumption that the deviations of spherical lensesare stochastically distributed.

8. Measurement Without the Steel Ball Using theDoubly-Reflected Wave of the Paraboloid

At first glance, one might assume that the test geo-metry can be simplified by exploiting the light doublyreflected from the parabolic mirror. But a closer ex-amination shows that this test does not reveal a un-ique profile of the mirror. In addition, the lightintensity of the probing wavefront is strongly de-formed in radial direction, since the incoming planewave is folded up in such a way that an inner annulusis inverted into an outer one and vice versa, stronglyincreasing the intensity toward the central region.

Fig. 17. Interferometric evaluation of the deviations of the used plane Fizeau reference surface against a supersmooth Fabry–Perot platewith λt=50 (all values are in waves).

Fig. 18. Repeatability test: Difference of the Zernike evaluationsof two measurements performed immediately one after the other(all values are in waves).

Fig. 19. Difference of the Zernike evaluations of two measure-ments with readjustment of the field selecting mask (all valuesare in waves).


This leads to problems with the dynamic range of theCCD-array, giving rise to systematic errors(see Fig. 20).Quantitatively, the measurement using the dou-

bly-reflected light in a parabolic mirror has to takeinto account that the height r of the incident rayand the height r0 of the doubly-reflected ray fulfillthe equation [1]

rr0 ¼ 4f 2 ⇒ r0 ¼ 4f 2

r: ð3Þ

By using the energy conservation in small annuliwith thickness dr and dr0, respectively, it is easy toshow that the intensity I0 of the doubly-reflectedwave and the intensity I of the incident wave arerelated by

Ij2πrdrj ¼ I0j2πr0dr0j ⇒ I0�r0 ¼ 4f 2

r

�¼

�r2f

�4IðrÞ:

ð4Þ

So, in our case of a focal length of f ¼ 2:1mm and amaximum radius of r ¼ rmax ¼ 10mm, the intensityin the center of the measurable area [with rmin ¼ r0 ¼1:764mm using Eq. (3)] would be I0ðrmin ¼1:764mmÞ ≈ 32Iðrmax ¼ 10mmÞ. Vice versa, at therim of the measurable area, it would be I0ðrmax ¼10mmÞ ≈ Iðrmin ¼ 1:764mmÞ=32. So, by assuming aconstant intensity of the incident wave (which isnot exactly the case when using radial polarizationbut shall be assumed here for the sake of simplicity)the dynamic range in the doubly-reflected wavewould vary by a factor of 1000.

Concerning the aberrations, a measurement usingthe doubly-reflected wave would give

Wtotalðr0;φ0Þ ¼ Wðr;φÞ þWðr0;φ0Þ

¼ Wðr;φÞ þW

�4f 2

r;φþ π

�

¼ W

�4f 2

r0;φ0 þ π

�þWðr0;φ0Þ: ð5Þ

Fig. 20. Interference pattern of the doubly-reflected plane wave and corresponding phase shift evaluations: Upper part, intensity dis-tribution seen by the CCD-camera; lower part/left, wrapped phase; lower part/right, phase distribution showing, in a qualitative manner,the threefold symmetry of the surface under test (all values are in waves).


Here, W are the surface deviations of the parabolicmirror in double pass, and ðr;φÞ and ðr0;φ0Þ are thepolar coordinates of a point of the incident and dou-bly-reflected wave, respectively. Of course, the polarangles are connected by φ0 ¼ φþ π. So, there is aquite complex mixing of the aberrations using thedoubly-reflected wave.

9. Discussion of the Experimental Results

In Section 7 the systematic error sources were dis-cussed to some extent. With this background thequality of the measuring results of the paraboloidtest shall be discussed focusing on the repeatabilityand reproducibility of the test method. Repeatabilityof phase shifting interferometry can be demon-strated by difference measurements of two runs car-ried out in the same adjustment and in closesuccession. This gives a repeatability of 0:025λt P/V and 0:002λt rms.From ray-trace simulations it can be concluded

that already 1 μm displacement of the ball spherecauses several wavelengths of aberrations. There-fore, it has to be expected that placement of the aper-ture selecting mask is very critical, which can beinferred from Fig. 19 where a small but inevitabledisplacement of the mask occurs if the interactiveplacement for the second run is carried out afterthe first run has been done. The wave aberration dif-ference is typically of the order of 0:06 − 0:08λt P/V.This is mainly due to the strong surface deviation,especially in the central region of the paraboloid.While the influence of the plane reference surface

onto the achievable accuracy can be neglected (seeFig. 17), this is not true for the steel ball referencesphere. Spherical surfaces with a 3:4π coverage ofthe solid angle have, so far, not been used as an ab-solute reference, which means that there is no guar-antee for error-free measurements with such anormal surface. However, there is the possibility toreduce the error probability through the followingtests: (1) it is possible to measure the paraboloid withdifferent steel balls and (2) to compare different mea-suring results after a rotation of the steel ball aroundthe axis of symmetry. Both possibilities have beenrealized and show deviations of the order λt=20 P/V (see Figs. 21–23, also Table 1).A hard test for measuring accuracy is a reproduci-

bility test with a time lapse of 4 weeks because thisincludes a new adjustment of the whole setup. In thiscase all possible error sources will contribute to thedifference of twomeasurements (see Fig. 24). The lat-eral displacement sensitivity can be demonstrated ifthe difference of the aberration data of the parabolicmirror is taken, i.e., the data are first displaced byseveral pixels and then subtracted from the originaldata. This will give the order of magnitude for thelateral displacement originating either from align-ment or from placing the data selection mask via in-teractive software. Since the Zernike expansion isbound to the software mask, the real data field mightdiffer from the center of the mask, causing systema-

tic aberrations. Figure 25 shows a possible differencefield of the wave aberrations caused by the parabo-loid with a shift matched to the displacement sus-pected to be present in the data of Fig. 24, left.The agreement in shape of Fig. 25 with Fig. 24, espe-cially with Fig. 24, left, indicates that the dominatingpart of the aberrations of Fig. 24 indeed stems from adisplacement of the data fields. Consequently, inFig. 26, where the data fields have been shifted withrespect to each other to eliminate these systematicaberrations in the best possible way, the P/V valuesare only half of their initial value.

Surface profile data can be calculated by taking thelocal angle of incidence into account, which dependson the entrance height of a beam onto the parabolicmirror surface. Since the surface is hit twice on theway to and from the reference sphere, it is necessaryto multiply the measured optical path length by afactor of gðθÞ ¼ 1=ð4 cos θÞ to obtain the surface devia-tions. Here, θ is the angle of incidence onto the localsurface that is connected to height r of the incidentray and focal length f by

r ¼ 2f tan θ ⇒ cos θ ¼ cos�arctan

�r2f

��

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�r2=4f 2

�s ⇒ gðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�r2=4f 2

�s

4: ð6Þ

Such an aperture-dependent renormalization forthe surface deviations is given in Fig. 27. The surface

Fig. 21. Difference of the Zernike evaluated wave aberrationsmeasured with two different reference steel spheres (all valuesare in waves).

Fig. 22. Difference of the Zernike evaluated wave aberrations oftwo consecutive runs, where the second run was carried out after a180° rotation of the steel sphere around the optical axis of theparaboloid (all values are in waves).


deviations are assumed as elevations or depressionsalong the surface normal and are defined positive inthe direction of the outward pointing surface normal.The angle-dependent phase shift introduced by thereflection at the metallic surface has been neglected.

10. Simulation of the Electric Energy Density in theFocus of the Paraboloid using the MeasuredAberrations

As mentioned above, the application of this parabolicmirror is to produce a nearly dipolelike radiant in-tensity to study the absorption probability of a singlephoton by a single atom or ion [2]. The transition ofthe 174Yb2þ ion at a wavelength of λe ¼ 252nmwill beused in this case. To get the influence of the aberra-tions onto the total electric energy density in the fo-cal plane of the parabolic mirror, a simulation usingthe Debye method in vectorial form is used [20]. Themeasured aberrations are scaled to a wavelength of252nm, i.e., the aberrations of Fig. 10 are multipliedwith a factor 633nm=252nm ¼ 2:51, and the aberra-tions are added to the phase of the incident planewave to simulate the influence of these aberrations.In the simulation, the amplitude distribution of theincident plane wave is assumed to be of such a kindnecessary to form an ideal dipolelike wave. The po-larization state is radially polarized. Figure 28 showsthe simulated square of the electric field, assuming atotal incident light power of 1W for the ideal casewithout aberrations and the actual case with aberra-

tions. Once more, the influence of angle-dependentphase shift upon reflection has been neglected.

It can be seen that themeasured aberrations are sohigh that the maximum value of the square of theelectric field in the case of aberrations is smallerby a factor of about 27 compared with the ideal casewithout aberrations. In the focus with aberrations,the trifoil symmetry of the measured aberrationsis also visible. So, for our final application, the errorsof the parabolic mirror have to be corrected. Since di-rect correction of the errors on the surface will not bepossible in practice, we will use a phase plate in frontof the mirror which compensates for the errors.

A diffraction limited optical system should have aStrehl intensity around 80% of the ideal system. Asimulation of the focus intensity using a symmetricalwave aberration, measured with the help of ourFizeau but limited to a P/V value of λe=4 (forλe ¼ 252nm) instead of 1:7λe P/V for the actuallymeasured aberrations, would result in 89% Strehl in-tensity (see Fig. 29). This delivers the order of mag-nitude for the allowable aberrations of the parabolicmirror in combination with the envisioned phasecorrection plate.

Table 1. Summary of Test Results

Type of Test Peak/Valley (λt) rms (λt) Comment

Repeatability (Fig. 18) 0.025 0.002 Just one run after the other, represents theaccuracy of the interferometer

Rotation 90° (Fig. 23) 0.045 0.003 Only steel sphere movedRotation 180° (Fig. 22) 0.039 0.002 Only steel sphere movedSelection mask readjusted (Fig. 19) 0.083 0.008 Only software manipulation, interactiveTilt versus fluffed out fringes (Fig. 16) 0.104 0.012 Plane reference mirror readjustedDifferent steel spheres (Fig. 21) 0.073 0.008 Whole setup readjustedReproducibility (Fig. 24) 0.106 0.013 Difference between runs 1 month apart, two different

adjustments, a lateral shift of the data setsresponsible for about 0:05 λ is at least probable

0.156 0.022

Reproducibility, adjusted for a lateralshift of the data fields (Fig. 26)

0.055 0.010 Suspected lateral shift of data fieldcompensated by software0.085 0.015

Fig. 23. Difference of the Zernike evaluated wave aberrations oftwo consecutive runs, where the second run was carried out after a90° rotation of the steel sphere around the optical axis of the para-boloid (all values are in waves).

Fig. 24. Reproducibility test with runs taken frommeasurementsdone with a time lapse of about 4 weeks, two different adjustments.Please note the systematic contribution of antisymmetric aberra-tion, whose cause has to be attributed to a lateral shift of the twodata sets measured with 4 weeks time lapse. It is obvious from thepeak/valley values that a bigger P/V value is connected with thedominant aberration features of a lateral displacement (comparethe left and right picture) (all values are in waves).


11. Conclusions

In this work we described the interferometric test ofa deep parabolic mirror to be used in an experimentalstudy of absorption and emission processes of singlephotons by single atoms. Experimental requirementsdemand the paraboloid to comprise a solid angle ofalmost 4π, with an overall surface quality of the orderof λe=10, where λe ¼ 252nm is the wavelength ofthe planned single photon absorption experiment.Regarding the test wavelength λt of 633nm of the in-terferometric setup, the accuracy requirementstranslate into a surface quality of at least λt=25.The steepness of the paraboloid makes the interfero-metric testing highly involved. However, the repeat-ability and reproducibility of the results show thatthe test can be carried out with the required accu-

racy, and the results with different steel ball orienta-tions show a sufficient agreement indicating that thesystematic error budget of the interferometer is al-ready small enough. The systematic measurementerrors may even be further diminished if measure-ments with the steel ball in different orientationsare taken and averaged; consequently, the deviationsof the steel ball almost cancel during the process.Since the steel ball can only be rotated around theoptical axis, a cancellation of its rotationally invar-iant deviations can be achieved by averaging overa set of different balls.

The measurement results show that the parabolicmirror under test has a surface quality of 0:4λt at thetesting wavelength, violating the λt=25 condition. Inthe final experimental setup, the wave aberrationsintroduced by the paraboloid, therefore, have to becompensated by other means. A possible way is touse a phase plate, analogous to the well-knownSchmidt plate in the case of spherical aberration,with a phase function creating the phase conjugateof the wave aberrations of the paraboloid. The sys-tem phase plate/parabolic mirror then acts as an al-most perfect parabolic mirror, enabling a successfulexecution of the final absorption experiment.

The authors express their gratitude to Lothar Dei-senroth of Carl Zeiss MicroImaging GmbHGöttingenfor providing the reference steel ball on top of a con-ical 1mm steel rod to enable the use of the steelsphere reference in our interferometric test. Theauthors also thank Fraunhofer IOF, Jena for manu-facturing the parabolic mirror.

Fig. 26. Reproducibility tests of Fig. 25, data fields laterallyshifted with respect to each other to demonstrate lateral displace-ment sensitivity: Left, difference of data fields of Fig. 24, left (oneof them shifted by 4 pixels in x and 2 pixels in y direction); right,difference of data fields of Fig. 25, right (one of them shifted by 7pixels in x and 4 pixels in y direction). In both cases the aberrationfeatures indicating lateral displacement of the fields have van-ished; the peak/valley values are almost halved (all values arein waves).

Fig. 27. Surface deviations of the paraboloid normalized via soft-ware from the measured wave aberrations (all values are inwaves).

Fig. 28. Square of the total electric field in the focal plane of theparabolic mirror by assuming, in both cases, a total incident lightpower of 1W: Left, ideal case without aberrations; right, actualcase for our mirror having the measured aberrations of Fig. 11.The maximum value of the square of the electric field is smallerby a factor of about 27.

Fig. 25. Difference of wave aberrations of the paraboloid after ashift of 4 pixels along x and 2 pixels along y direction, see Fig. 24,left as well as Fig. 27, left (all values are in waves).


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