Reconstruction of an object shape from thenear-field intensity of a reflected paraxial beam

Gleb Vdovin

Experimental reconstruction of an object shape from the near-field intensity of reflected or transmittedlight is reported. The method of reconstruction is based on the direct numerical solution of the finite-difference representation of the paraxial irradiance-transport equation. Practical applications and lim-itations of the method are discussed. © 1997 Optical Society of America

1. Formal Background

Reconstruction of the phase of a paraxial light beamis a traditional problem of applied optics. Methodsthat are based on direct intensity measurementsare of special interest, as in many cases the onlyavailable information consists of the measured in-tensity. This group includes Hartmann-basedmethods,1,2 methods based on phase reconstructionwith iterative transform algorithms,3,4 and meth-ods that use the irradiance-transport equation.5–8

The above-mentioned approaches were developed todeal with problems of ~real-time! wave-front recon-struction in adaptive optics and precision opticaltesting. In this paper the irradiance-transportequation is used to reconstruct wave-front aberra-tions on a comparatively large scale by use of theregistered intensity distribution of light that wasreflected or transmitted by an object. The phase isobtained as a solution of a system of linear equa-tions, converting the irradiance transform equationinto its finite-difference analog. The method canbe used for preliminary optical testing of refractiveand reflective objects that have smooth asphericalshapes with relatively large deviations from thereference plane or sphere.

Let us consider the wave function U that conformsto the time-independent wave equation,

DU 1 k2U 5 0, (1)

The author is with the Laboratory of Electronic Instrumenta-tion, Delft University of Technology, P.O. Box 5031, 2600 GA,Delft, The Netherlands.

Received 13 June 1996; revised manuscript received 10 January1997.

0003-6935y97y225508-06$10.00y0© 1997 Optical Society of America

5508 APPLIED OPTICS y Vol. 36, No. 22 y 1 August 1997

where D 5 ]2ydx2 1 ]2ydy2 1 ]2ydz2. The solution ofEq. ~1! in the paraxial approximation has the formU~x, y, z! 5 A~x, y, z!exp~ikz!; substituting this intoEq. ~1!, we obtain the equation for the slowly chang-ing amplitude of the electromagnetic field A~x, y, z!:

D'A 1 2ikdAdz

5 0, (2)

where D' 5 ]2y]x2 1 ]2y]y2. Assuming that A~x, y,z! 5 @I~x, y, z!#1y2exp@iw~x, y!#, where I is the inten-sity, we derive the equation that describes the trans-port of irradiance in the paraxial approximation,

ID'~w! 1]I]x

]w

]x1

]I]y

]w

]y1 k

dIdz

5 0. (3)

In the analysis below Eq. ~3! is applied to recon-struct the phase from the near-field scattered inten-sity. It was shown in Ref. 7 that Eq. ~2! correctlydescribes the near-field propagation of an aberratedlight beam if these conditions hold:

Q2 5 ~nlya!2 ,, 1, a .. l. (4)

Here Q is the maximum local wave-front tilt intro-duced by the aberration, nl is the amplitude of theaberration, and a is its transverse scale. These con-ditions require the aberration to be smooth, but donot impose any limitations on its amplitude.

The irradiance-transport equation ~3!, with appro-priate boundary conditions, can be used to recon-struct shapes of reflective or refractive objects by useof the experimental setup shown in Fig. 1.

The experiment consists of recording two intensitydistributions projected onto the remote screen: Thefirst, I~x, y, 0!, is recorded by the focusing of the lensonto the surface of the object; the second, I~x, y, Z!, isrecorded after the lens slides to distance Z along the

reflected beam. If the condition L .. f .. Z is sat-isfied ~where L is the initial distance between the lensand the screen, Z is the shift of the lens and f is thefocal distance!, the shift of the defocused imagedplane can be set equal to Z and the change of theimage scale in the plane of the screen can be ne-glected. Otherwise the defocused image and effec-tive lens shift should both be additionally scaled by afactor of ~L 2 f !y~L 2 Z 2 f !.

In practice, condition ~4! is satisfied if all the lightreflected or refracted by the phase object ~which in-troduces no intensity modulation! is collected by alens with a small numerical aperture. In this case,the image of the phase does not have any intensitymodulation inside the object contour when the lens isfocused onto the object.

The profile of the object, which corresponds in thedescribed approximation to the phase of the reflectedlight, can be reconstructed from the recorded inten-sity distributions by the solution of the finite-difference analog of Eq. ~3! numerically withappropriate boundary conditions. One of many pos-sible finite-difference presentations of Eq. ~3! has theform

Ii, j~wi11, j 1 wi, j11 1 wi21, j 1 wi, j21 2 4wi, j!

d'2

1k~Ii, j

1 2 Ii, j!

dz1

~wi, j11 2 wi, j21!~Ii, j11 2 Ii, j21!

4d'2

1~wi11, j 2 wi21, j!~Ii11, j 2 Ii21, j!

4d'2 5 0, (5)

where the intensity I and the phase w are both de-fined in a uniformly sampled square grid with equalsteps d' in the x and the y directions and step Z 5 dzin the direction z, so that

Ii, j 5 I~id', jd', 0!,

wi, j 5 w~id', jd', 0!,

Ii, j1 5 I~id', jd', dz!.

By isolating wi, j in Eq. ~5!, we obtain

wi, j 5k~Ii, j

1 2 Ii, j!

4Ii, j1 dz

1wi11, j 1 wi21, j 1 wi, j11 1 wi, j21

4

1~wi11, j 2 wi21, j!~Ii11, j2Ii21, j!

16Ii, j1 d'

2

1~wi, j11 2 wi, j21!~Ii, j11 2 Ii, j21!

16Ii, j1 d'

2 . (6)

To solve the system of linear equations ~5!, expres-sion ~6! should be iterated over all indices i and j untilconvergence. The process starts from an arbitraryphase distribution, for example w 5 0. Applicationof a direct method for solving the system of equations~5! requires computer memory storage of a matrix ofN4 coefficients for a problem involving an image withN2 pixels ~108 coefficients for an image of 100 3 100

pixels!. The direct solution is not always justified,even if special methods of sparse storage are used.

The proposed method has a clear advantage ofhigher transversal resolution over Hartmann-likemethods.1,2 In the proposed method, the samplingof the input intensity array is equal to the samplingof the reconstructed output phase array. In theHartmann method the phase tilts are sampled over araster of localized subapertures, and in theHartmann–Shack method the phase tilts are aver-aged over subapertures of the raster. In both vari-ants the resolution is limited by the input sparseraster of subapertures.

The described method has an advantage over fastFourier-transform methods,9–11 which reconstructthe phase as an argument of a complex variablewrapped into the interval @0, . . . , 2p#. The proce-dure of unwrapping is numerically unstable and can-not be performed for quickly changing or noisy phasefunctions. The finite-difference method reconstructsthe phase as a real function, eliminating 2p unwrap-ping and allowing for correct reconstruction of largedeviations in the reconstructed profile.

Fast Fourier-transform methods impose artificialperiodic boundary conditions along the borders of thecalculation region. The finite-difference representa-tion @Eq. ~3!# allows, in principle, the use of any math-ematically valid explicitly defined boundarycondition. This is a serious advantage that extendsthe applicability of irradiance-transport methods.Setting ID'~w! 5 0 in Eq. ~3!, which is true at theobject boundary where the local curvature equalszero, we obtain the expression for Neumann bound-ary conditions similar to that obtained in Ref. 12:

]I]x

]w

]x1

]I]y

]w

]y1 k

dIdz

5 0. (7)

Expression ~7! can be uniformly converted into afinite-difference presentation by use of the approachdescribed above. The phase of the reflected light isreconstructed by the solutions of Eq. ~3! inside thecontour and Eq. ~7! for points belonging to the explic-itly defined contour. As the intensity changesquickly in the contour region, the precision of themethod depends on the precision of the contour-pathdefinition: Small contour deviations cause signifi-cant errors of reconstruction. Thus methods thatexclude reconstruction of the boundary conditions

Fig. 1. Experimental setup used to reconstruct the shape of a softreflective phase object. For a transparent refractive object, thecollimated beam should be transmitted by the object.

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from recorded intensities are of interest. For in-stance, zero boundary conditions for the phase w 5 0can be explicitly defined along the boundary of a largearea surrounding the registered intensity. In thiscase Eq. ~3! is solved for all points of the illuminatedarea; otherwise the Laplace equation D'w 5 0 issolved for the rest of the surrounding area. If thedistance between the object contour and the contourof the artificially added surrounding is large enough~larger than the object size!, the influence of the in-troduced artificial boundary is small and the recon-structed shape corresponds approximately to theshape of the object.

Both methods of treating boundary conditions havetheir drawbacks that reduce the robustness of thereconstruction. Any a priori information about theshape of the object, and especially about the objectcontour, must be taken into account at the recon-struction stage, as it will greatly improve the robust-ness and the precision of the result.

Finally, as to the relation of the proposed method tothe group of iterative transform methods of phasereconstruction, we can state that the considered con-figuration allows for a direct one-step reconstructionof the phase function. Such a simplification is pos-sible because the method is restricted to only the caseof near-field propagation of a paraxial beam, which isdescribed correctly by a one-step linear finite-difference approximation of the irradiance-transportequation.

2. Experiment

In the experiment, the shape of a defect membraneadaptive mirror,13 which had light aperture of 9 mm,was reconstructed. Because of an error that oc-curred during the packaging stage of mirror fabrica-tion, the space between the reflective membrane andthe control electrode structure was partially filledwith liquid epoxy. After polymerization, the ini-tially plane membrane was strongly distorted by acombination of capillary forces and epoxy shrinkage.Attempts to measure its shape interferometricallyfailed because of the presence of a strong asymmetricaberration. A mechanical profilometer could not beused because it could break the freely suspendedparts of the 0.7-mm-thick membrane. Thus themethod based on the irradiance-transport equationwas used.

Two intensity distributions ~see Fig. 2! were regis-tered with the setup shown in Fig. 1; the lens had afocal distance of f 5 150 mm and an aperture of 2a 525 mm. The difference between two intensity distri-butions is enhanced. In practice, the difference be-tween two images is hardly visible because the linearapproximation of the irradiance-transport equationis valid only in the case of a weak intensity modula-tion.

The reconstructed shape of the wave front, re-flected on the deformed membrane, is shown in Fig. 3.

The maximum amplitude of the reconstructedphase equals 1560 rad, which corresponds to 78-mmmembrane deformation. This value is in good agree-

5510 APPLIED OPTICS y Vol. 36, No. 22 y 1 August 1997

ment with direct measurements, which were con-ducted by the scanning of the object with amicroscope, focused onto its surface: Fig. 4 shows acomparison between the direct measurement and thereconstruction. The reconstructed phase changesfrom 0 to 21560 rad in the interval, including only 35sampling points, which corresponds to a speed ofphase change of at least 45 rad per sampling interval.To reconstruct the same profile with any method re-quiring phase unwrapping, a grid with a density thatis at least 30 times higher ~1.5 rad per samplinginterval! should be used to secure successful unwrap-

Fig. 2. Intensity distributions formed by the lens, focused on theobject surface ~left! and slid 5 mm along the beam ~right!.

Fig. 3. Reconstructed shape of the reflected wave front. Recon-struction was conducted on a rough grid of 100 3 100 samples.

Fig. 4. Mirror profile measured by the scanning microscope~dashed curve! and reconstructed from intensity measurements~solid curve!.

ping. This would require the input intensity to beregistered in a grid of 9000 3 9000 samples and 900times more memory for processing the image.

The described method relies on precise intensityregistration. For a soft phase object, the secondterm of Eq. ~3!, which describes the intensity modu-lation that is due to local tilts, is small compared withthe first curvature term ~no intensity modulation inthe focused image!. The relation

ID'F 5 1y~2Rl! <I 2 I1

Idz, (8)

where F 5 kw is the shape of the phase object and Rlis the local radius of curvature of the reflected ortransmitted wave front, illustrates the influence ofsystematic errors of measurement of intensities I andI1. These errors introduce an additional parabolicterm into the reconstructed phase. The simple cri-terion of intensity conservation,

*S

IdS 5 *S

I1dS, (9)

where S is the image area, must hold, otherwise thesystematic parabolic error is superimposed onto thereconstructed shape.

Nonlinearity of the registration leads to distortionsof the reconstructed profile. In the simplest case,the degree of nonlinearity of the image recording sys-tem is described by the parameter g. The measuredintensity Im depends nonlinearly on the incidenceintensity I:

Im 5 Imax~IyImax!1yg; (10)

thus if g , 1 the contrast of bright details is improvedwhile the contrast in the shadows is reduced; g . 1reduces the contrast of bright details and improvesthe contrast in the dark regions.

To analyze the influence of the g factor on the qual-ity of the reconstruction, corrections with differentvalues of g have been applied to the originally regis-tered intensity distributions shown in Fig. 2. Theresultant phase profiles, reconstructed from the cor-rected distributions, were compared with the origi-nal, corresponding to g 5 1. The rms error of thereconstruction and the peak-to-valley amplitude ofthe reconstructed profile are shown in Fig. 5 as func-tions of g.

The influence of the nonlinearity of the photore-ceiver is illustrated by Fig. 6. For a wave front thatcomprises convex and concave regions, the amplitudein concave regions increases during the propagationwhile the amplitude in convex regions decreases. Ifg . 1, the positive modulation in concave regions isreduced and the negative modulation in convex re-gions is amplified by the nonlinearity. This leads toan overestimation of the amplitude of convex defor-mations in the reconstructed profile and, appropri-ately, underestimation of the amplitude of concavedeformations. If g , 1, the result is the opposite.

In the present experiment the measured profile wasconcave; therefore increasing g reduced the overallamplitude of the reconstructed deviation from plane.

To analyze the influence of noise on the precision ofreconstruction, additive random noise with a uniformdistribution was added to the originally registeredimages. The influence of image noise is illustratedin Fig. 7. Additive random noise significantly in-creases the intensity in dark regions of the image,while the bright regions are left almost unaffected.Thus the effect is similar to a positive g correction,resulting in amplification of convex deformations andunderestimation of concave amplitudes in the recon-structed profile. As the test profile is concave, theamplitude of the reconstructed profile decreases withan increase in the noise component.

The precision of the method also depends on thedepth of the observed amplitude modulation. Re-constructions from weakly modulated images arenoise sensitive, whereas too strong modulation is notcorrectly described with the developed model of theone-step first-order finite-difference approximation.

Similar experiments have been conducted withvarious reflecting objects; for example, the shape ofthe blade of a Swiss army knife was reconstructed bythe measurement of two reflected intensity distribu-

Fig. 5. Peak-to-valley amplitude of the reconstructed profile andthe rms error as functions of g.

Fig. 6. Nonlinear intensity conversion for different values of theparameter g.

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tions. The intensity distributions and the recon-structed solid model are shown in Fig. 8. In thisparticular case the condition of a soft phase object isnot fully satisfied, as the light is also scattered bysurface scratches. The boundary conditions cannotbe extracted from registered intensities because ofthe bad optical quality of sharp and rough knife

Fig. 7. Root-mean-square error of the reconstruction and the am-plitude of the reconstructed profile as functions of noise compo-nents in registered intensity distributions.

Fig. 8. Two images of the tip of a Swiss army knife and thereconstructed solid model.

5512 APPLIED OPTICS y Vol. 36, No. 22 y 1 August 1997

edges. The method of remote zero boundary condi-tions, as described above, was used. The rms errorof reconstruction, estimated by a comparison of sec-tions of the reconstructed profile with direct profilemeasurements, was better than 20%; the peak-to-valley amplitude of the reconstruction was approxi-mately 40% larger than that measured directly, withthe maximum deviation lying in the shadow regionalong the boundary of the illuminated area. Theseresults illustrate the expected error magnitude forscattering objects with badly defined boundary con-ditions. In spite of the substantial error, the recon-struction provides a realistic description of the objectshape and allows for estimation of important geomet-ric parameters, such as surface curvature.

3. Conclusions

The conducted experiments show that the procedureof wave front reconstruction, based on finite-difference approximation of the irradiance transportequation, is applicable for the reconstruction of wavefronts, scattered on soft specular reflective surfaces,or refractive objects with relatively large deviationsfrom plane. These profiles in many cases are diffi-cult for reconstruction with traditional interferomet-ric methods or Fourier-based irradiance-transportmethods. The described method is applicable to awide range of problems of optical metrology, re-stricted to the paraxial-beam approximation. Themethod is sensitive to systematic errors, nonlinear-ity, and noise in registered intensity distributions.Experiments and numeric simulations show that ifthe sum of systematic errors, nonlinearity, and noiseis of the order of 1%, the error of reconstruction is nothigher than 5% for experiments with correctly de-fined boundary conditions. The rms error mayreach 10%–50% for noisy and nonlinear measure-ments and for objects with optical defects and poorlydefined boundary conditions. Even in these cases,reconstructed functions will give a good impression ofthe shapes of the examined objects.

This work has been partially supported by the long-term EC European Strategic Programme for R&D inInformation Technology project 21063, Micro-OpticalSilicon Systems.

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