Three-dimensional sensing by using alenslet, a Dammann grating, and a combination

David Mendlovic and Eran Gur

In the last two decades, three-dimensional sensors based on misfocusing have been suggested. Thisresearch addresses the question of measuring several focal depths simultaneously. Two options forgenerating the necessary array of spots are analyzed: the use of a lenslet array and the use of aDammann grating. Finally, a combination of the two approaches is proposed. Such a combinationenables tailoring the system performance to the exact needs of the user. © 1998 Optical Society ofAmerica

OCIS codes: 150.6910, 050.2770, 040.1240.

1. Introduction

Three-dimensional surface sensing is an intensiveresearch issue. It seems that in many cases knowl-edge of the two-dimensional image structure is insuf-ficient and does not contain enough informationregarding the structure of the investigated objects.A good example is the advance in products of micro-electronic devices and surface relief diffractive opticalelements. In these items, a two-dimensional topview is insufficient for discovering small flaws and athree-dimensional measure is needed.

Fainman et al.1 suggested 15 years ago an opticalsetup for measuring surface topology based on mis-focusing. In their setup a beam reflected from atested object was split into two separate arms. Thetwo beams passed through filters placed at precalcu-lated positions, and the difference between the out-puts of these filters was fed into a translator. Thetranslator shifted the object forward or backwardalong the beam propagation direction, closing a feed-back system. As can be easily understood, the exactpositioning of the filters is critical for obtaining a goodperformance from the setup, and for every singlepoint on the object, a feedback procedure is needed.Thus the whole process is slow.

More recently, a method for measuring surface to-

The authors are with the Department of Physical Electronics,Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Is-rael.

Received 2 July 1997; revised manuscript received 18 September1997.

0003-6935y98y010125-05$10.00y0© 1998 Optical Society of America

pology that uses a lens with high chromatic aberra-tions was proposed by Hutley and Stevens.2 Withthis system, a Fresnel-zone-plate lens was used basedon its well-known dispersive behavior. A pinholecaptured the center of the main lobe whose wave-length was measured with a spectrometer device.Later, this method was simplified by using a lookuptable that associates each depth with the differentwavelength measurement of a color CCD camera.3This modification accelerated the measurement pro-cess significantly. Full surface mapping wasachieved by replacing a single Fresnel zone platewith an array of N 3 N Fresnel zone plates. Then,with the color technique of the same lookup table, thedepth of each N 3 N measured point was found.The simultaneous measurement of 100 depths wasdemonstrated with this method.

The aim of this project is to analyze various optionsfor generating an array of spots that will be usedbelow for large-surface multiple-depth sensing. Thedepth sensing can be done by using either the misfo-cus approach or the color sensing approach.

The first method, for generating an array of spots~Section 2!, is based on a lenslet array. Here thethree-dimensional measurement is limited accordingto the number of lenses. If a small number of large-aperture lenses are used, broadening to the depthratio ~numerical aperture! of the beam will be smalland thus the depth resolution will be high, but thespatial resolution will be small. If a large number ofsmall-aperture lenses is used, the spatial resolutionincreases and the depth resolution decreases.

The second approach ~Section 3! is based on a Dam-mann grating as a spot array generator that uses theentire aperture of the input beam.4 Here the beam

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widens quickly before and after the focal plane, andeach small change in the depth causes a large changein the detectors. Thus very rough surfaces cannotbe mapped. The spatial resolution in this case islimited by the number of reproductions that the Dam-mann grating generates. This number depends ontechnological limitations.

Both approaches require special care to overcomethe depth redundancy due to the symmetry beforeand after the focal plane. This is done in Section 2.In Section 4 we propose a combination of these twoapproaches. Such a combination might be useful fortailoring a system to a very-well-defined perfor-mance. In the combined setup, a lenslet array ofmany mid-size lenses is used, thus increasing spatialresolution. Each lens produces a subarray of spotswhen a Dammann grating is attached to a mid-sizelens. One can design the exact spatial and depth offocus resolutions to be performed by the system bycontrolling the lens aperture and the order of theDammann grating, respectively.

2. Array of Depth Sensing in which a Lenslet Arrayis Used

The use of the focusing behavior of a lens for surfacemeasuring was suggested by Fainman et al.1 in 1982.A simplified variation of this system is as follows. Aplane wave passes through a lens, and the reflectedconverging beam propagates a distance relative tothe depth of the tested surface before entering thedetector. Because of the misfocus, the size of thespots on the detector is proportional to the depth ofthe object at the measured location. Thus smallchanges in depth yield small changes in beamwidth.We propose here that this idea be used for an array ofdepth sensing by replacing the focusing lens with alenslet array. This setup is shown in Fig. 1.

Because of the symmetry before and after the focalplane, there is a depth redundancy in the method.As can be seen from Fig. 1, the spot sizes for beams ~a!and ~b! are the same, although the depths are differ-ent, since one beam hits the surface before its focalplane and the other right after its focal plane. Toovercome this redundancy, one may implant anasymmetrical defect in each of the lenses ~the samedefect for all lenses!. Thus in the detector we willwitness a similar defect. According to the defect po-

Fig. 1. Setup based on a lenslet array used for depth sensing.

126 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

sition ~relative to the optical axis!, we will be able todetermine whether the beam hits the surface beforeor after the focal plane. If detection is done with aCCD camera, there is enough knowledge regardingthe structure of the spot for such a decision.

An inexpensive way to implement the lenslet arrayis with a Fresnel-zone-plate array. There such adefect can be generated as shown in Fig. 2. Becauseof the defect shown in Fig. 2~a!, a high-frequencysignal is generated, and in the Fourier domain it istranslated to background noise around the main peakas demonstrated in Fig. 2~b!. If, however, we look atthe spot just before the focal plane we can see that thenoise is shifted toward the direction of the defect andthus the peak is no longer in the center of the mainlobe. This is evident from Fig. 2~c!. Figure 2~d!demonstrates the shape of the spot, generated by thedefective zone plate, behind the focal plane. Onecan see that the peak shifts slightly along the otherdirection.

A hand-waving explanation might be as follows.For a simplified mathematical analysis, we concen-trate on the one-dimensional case, the x-axis defect.If an incident plane wave passes through an ideallens, in the Fourier plane a delta function ~plus qua-dratic phase! will be generated in the focal plane.Multiplying the input by an asymmetrical rectangu-lar function, rect@~x 1 dy2!yd#, yields in the Fourierplane a convolution between the delta function and asinc function with a linear phase. This leads to asinc amplitude distribution multiplied by a linearphase around the center of the Fourier plane:

sincSdn

lfDexpS2 ipdn

lf D,

where n is the coordinate of the Fourier domain andf is the lens focal length.

Advancing an extra distance z behind the focalplane, according to the free-space propagation rule~Kirchhoff integral!, the linear phase shifts the beamto a new spot position. The lateral shift of the spotis @z~dy2f !#. For a positive value of z ~in front of thefocal plane!, the peak of the sinc will shift to theposition opposite the defect in the lens. A negativevalue of z yields the opposite shift.

If the lenslet array consists of N 3 N lenses with a

Fig. 2. ~a! Defect in the zone plate. ~b! View of the defect in theFourier ~focus! plane. ~c!, ~d! View of the defect in the forward andbackward out-of-focus plane, respectively.

total aperture size P and a focal length f, the width ofthe beam placed a distance Z behind ~or before! thefocal plane will be ~when a geometric optics approx-imation is used!

WBeam 5 ZPNf

(1)

as may be obtained from Fig. 3~a!.However, for large values of N, geometric optics

approximations are no longer valid and diffractiveoptics should be addressed. In this case the beambroadens as the replica number increases. Thus thereplica number must be limited. In a Gaussianbeam, the beam broadens according to Eq. ~2! @as canbe obtained from Fig. 3~b!#:

WBeam

Z5 uBeam 5

l

pW0. (2)

Thus the generation of many narrow spots will lead tofast undesirable broadening.

On the other hand, the spatial resolution of thelenslet array is limited by

D 5 PyN. (3)

With this method the three-dimensional measure-ment is limited either in the spatial resolution ~which

Fig. 3. ~a! Optical beam broadening according to the geometricoptics approximation. ~b! Optical beam broadening according todiffractive optics.

requires a higher P! or in the depth resolution ~whichrequires a high N!.

The notations mentioned in this section are alsoapplicable for the color-coding technique, as de-scribed above.

3. Array Sensing by Using a Dammann Grating

A second approach for performing array sensing isbased on the use of a single lens attached to a Dam-mann grating as demonstrated in Fig. 4. First, werecall the concept of the Dammann grating as notedby Dammann and Gortler in 1971.4 The Dammanngrating is a periodic binary phase grating where ev-ery period can have a highly complex structure. Ow-ing to the periodic structure, the grating functionmay be expressed mathematically as a Fourier series.In other words, the Fourier image of a wide Dam-mann grating is a sum of shifted delta functions.

The amplitudes of the delta functions are associ-ated with the Fourier series coefficients and are setaccording to the inner structure of the grating’s basicperiod. Consider the case of a two-level phase-onlygrating with M transitions from 2p to p for eachbasic period. In this case, one has the freedom ofwhere to place these M transitions. Different solu-tions will yield different Fourier coefficients in theFourier domain and different light efficiencies.When the grating is a Dammann grating, it will gen-erate a set of ~2M 1 1! 3 ~2M 1 1! uniform amplitudedelta functions in the Fourier plane as well as someside spots in the outer area.

Thus, if a beam passes through a lens and thenthrough a Dammann grating, in the lens focal planeit provides a set of uniform amplitude delta functionswith a quadratic phase. The beam will expand be-fore and after the focal plane as in any focusing setup,and this expansion is exactly the property measuredby the detector. If the lens aperture in this setup isequal to the total aperture used in the previous setup,one may express the misfocused beamwidth as afunction of Z by @obtained in a way similar to that ofEq. ~1!#

WBeam 5 ZPF

. (4)

Fig. 4. Dammann-grating-based setup for depth sensing: BS,beam splitter.

1 January 1998 y Vol. 37, No. 1 y APPLIED OPTICS 127

This equation means that the beam-converging angleis N times bigger than in the previous case. Thusevery small change in the medium topology causes arelatively large change in beamwidth, i.e., the depthresolution is N times larger than the one in the pre-vious setup but the total possible measured depth islimited. Note that the spatial resolution is limitedby the number of uniform amplitude spots generatedby the Dammann grating. This number is restrictedby the technology that generates the Dammann grat-ing5 that may be expressed by

D ImIm

5 8S2hD

1y2

N3y2 1Q

. (5)

In Eq. ~5! ~D Im!yIm is the maximal relative unifor-mity error of the intensities in the Fourier plane, h isthe light efficiency, Q is the ratio between the size ofone grating period and the plotter resolution, and Nis the number of the delta replica generated by theDammann grating in the Fourier plane. Using theup-to-date plotters, one cannot reach beyond a 25 325 uniform array. @For example, using a 0.25-mmplotter resolution to create a 1-mm period gratingwith efficiency higher than 75% and energy unifor-mity of at least 90% is possible only when an array ofless than 20 3 20 spots is generated, as can be seenfrom Eq. ~5!.#

As in the previous approach, the symmetry beforeand after the focal plane causes a depth redundancy.One may overcome the redundancy by inserting adefect into the large lens, similar to the defect dis-cussed in Section 2.

4. Combined Approach

Following the discussion provided in Sections 2 and3, it is clear that a lenslet-array-based sensor enablesonly the measurement of rough, more obvious surfacetopology. On the other hand, a Dammann-grating-based sensor enables the measurement of fine details,but it is too precise for measuring large depth differ-ences. In both setups the spatial resolution is lim-ited, either by the number of measuring points perarea ~in the Dammann-grating-based setup! or by thetrade-off between spatial and depth resolution ~in thelenslet-array-based setup!. This limitation moti-vates the combination of the two methods of surface

profiling for achieving a flexible ability to tailor thesystem to the exact requirements of the user.

The proposed combined setup is demonstrated inFig. 5. The incident light beam passes through alenslet array, where each lens is followed by a Dam-mann grating. If the lenses have small apertures,each of them handles a small fraction of the entiretested surface. In these small subareas, the changesin depth are expected to be relatively small. Thusthese subareas are moderate and can be mapped by aDammann-grating-based sensor. In such a setupthe surface can be fully scanned with sufficient accu-racy regardless of its topography.

Let us assume N 3 N lenses in the lenslet arrayand an M 3 M replica Dammann grating. The totalnumber of measured spots is ~NM! 3 ~NM!. Thiswill lead to the same beam broadening as in thesingle-Dammann-grating setup but the width of thebeam is N times smaller because of the aperture ofeach lens ~PyN, where P is the total aperture size!.

The Dammann grating yields, according to theFourier series theory, the amplitude distribution

D~n! 5 ( AmdS n

lf2

mdD (6)

near the focal plane @d 5 Py~NyM! is the gratingperiod#. Detection of the spots in the Fourier planedepends on their separation. Therefore, to preventaliasing, we demand that the width of a single beamhitting the detector not exceed the separation be-

Fig. 5. Combined lenslet-array–Dammann-grating setup.

Table 1. Energy Distribution in Different Diffraction Orders, Generated by Several Dammann Gratingsa

ReplicaNumber~2N 1 1!

Order of Diffraction Energy inUniform

Orders ~h!0 1 2 3 4 5 6 7 8

3 0.221 0.221 0.100 0.016 0.000 0.011 0.010 0.001 0.000 66.4%5 0.154 0.154 0.154 0.002 0.002 0.032 0.000 0.011 0.008 77.4%7 0.093 0.093 0.093 0.093 0.032 0.000 0.005 0.043 0.019 65.5%9 0.069 0.069 0.069 0.069 0.069 0.008 0.034 0.000 0.017 62.8%

11 0.059 0.059 0.059 0.059 0.059 0.059 0.000 0.014 0.013 64.9%15 0.042 0.042 0.042 0.042 0.042 0.042 0.042 0.042 0.001 63.1%

aThe boldface data are for the active orders, the rest of the data is for undesired orders.

128 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

tween two beams. Thus the limitation for surfacemeasuring is

Z #lf 2NPd

. (7)

This formula is based on demanding that the gapbetween two peaks Dn 5 nm11 2 nm @given by Eq. ~6!#exceed the width of the spot WBeam as given in Eq. ~4!.

Another issue that should be addressed is the uni-formity of the spots. Indeed the spots that are gen-erated by the Dammann grating are uniform.However, their sidelobes interfere with the sidelobespots generated by the nearby lenslet. This inter-ference is illustrated in the original work of Dam-mann and Gortler.4 An example of energydistribution is given in Table 1. As shown, increas-ing the number of uniform spots does not automati-cally increase the energy in the main lobes. Thus asignificant portion of the energy present in the side-lobes may cause some interference in the results.

Finally, we point out that the depth redundancyproblem still exists in the combined setup. The so-lution, as in the previous setups, is to generate anasymmetrical defect in each of the lenses affectingthe image on the detector.

5. Conclusions

We have described two methods of measuring a si-multaneous array of depths. Both methods arebased on out-of-focus sensing. The first setup uses alenslet array to generate an array of spots illuminat-ing the tested surface. The use of zone plates in thissetup enables color coding of the depth instead ofout-of-focus sensing. The second method uses aDammann grating for generating the array of spots.

It has been shown that the lenslet based setup isrecommended for rough surface measurement

whereas the Dammann-grating-based setup providesadvantages for detection of finer surfaces. From alateral-spatial-resolution point of view, the lenslet-based system is limited by the required depth reso-lution ~which defines the minimal lenslet aperture!and the Dammann-grating-based setup is limited bytechnological limitations.

We have suggested a combination of the abovemethods into a single optical setup. The combinedapproach allows depth measuring of high complexityand of large-surface objects. It offers full control ofthe array size, lateral spatial resolution, and depthresolution. In all setups the redundancy problemwas solved when an asymmetrical defect was gener-ated in the lens or lenslet array.

The authors acknowledge partial financial supportprovided by The Israeli Ministry of Science–DOE in-frastructure grant.

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3. D. Mendlovic, “Three-dimensional image sensing based on azone-plate array,” Opt. Commun. 95, 26–32 ~1993!.

4. H. Dammann and K. Gortler, “High-efficiency in-line multipleimaging by means of multiple phase holograms,” Opt. Commun.3, 312–315 ~1971!.

5. J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, and N. Streibl,“Dammann grating for laser beam shaping,” Opt. Eng. 28,1267–1275 ~1989!.

6. G. Molesini, G. Pedrini, P. Poggy, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49,229–233 ~1984!.

7. G. Hausler and D. Ritter, “Parallel three-dimensional sensingby color-coded triangulation,” Appl. Opt. 32, 7164–7169 ~1993!.

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