The final version of this preprint is available fromhttp://iopscience.iop.org/article/10.1088/2040-8978/17/12/122001

Lensless coherent imaging by sampling of the optical field with digital

micromirror device

G. Vdovin,1,2,3 H. Gong,1 O. Soloviev,1,3 P. Pozzi, 1 & M. Verhaegen1

1DCSC, TU Delft, Mekelweg 2, 2628CD, Delft, The Netherlands2ITMO University, Kronverksky 49, 197101 St Petersburg, Russia

3Flexible Optical BV, Polakweg 10-11, 2288GG Rijswijk, the Netherlands∗g.v.vdovine@tudelft.nl

October 21, 2016

Abstract

We have experimentally demonstrated a lensless coherent microscope based on direct registration ofthe complex optical field by sampling the pupil with a sequence of two-point interferometers formedby the digital micromirror device. Complete registration of the complex amplitude in the pupil ofimaging system, without any reference beam, provides a convenient link between the experimental andcomputational optics. Unlike other approaches to digital holography, our method does not require anyexternal reference beam, resulting in a simple and robust registration setup. Computer analysis of theexperimentally registered field allows for focusing the image in the whole range from zero to infinity,and for virtual correction of the aberrations present in the real optical system, by applying the adaptivewavefront corrections to its virtual model.

Keywords: Microscopy, Digital holography, Inverse problems, Phase retrieval, Adaptive optics

Reconstruction of the object by methods of dig-ital holography [1] requires the complex amplitudeof the field to be known with high spatial resolu-tion. Since direct registration of the optical com-plex amplitude is impossible, a number of indirecttechniques have been developed. Methods of dig-ital holography, including phase-shift interferome-try [2–4] require the reference beam to be presentin the system. Such reference beam can be in-ternally generated by spatial filtering of a portionof the object beam [5, 6], or through the use oflight from an empty area of the field of view [7].Reconstruction of phase from the intensity of thediffracted field [8–10] generally represents an ill-posed problem, and requires some additional in-formation about the object to be made available.Reconstruction of the wavefront without a refer-ence beam is possible with Shack-Hartmann sensor[11]. Recent publication [12] describes a quadriwavelateral shearing interferometer for high-resolutionfield reconstruction. However, methods based onthe phase reconstruction from wavefront tilts, usu-

ally reconstruct only the potential component of thephase, neglecting phase vortices. Also, a relativelyuniform intensity distributions needs to be presentin the plane of reconstruction. These two factorssignificanlty limit the applicability of these methodsto digital holography. Time multiplexing [13, 14]represents another useful approach to high resolu-tion reconstruction of the complex fields. In thisapproach the complex amplitude is registered se-quentially in different locations of the pupil. This isusually achieved through complex setups requiringa reference beam and a galvanometric or piezoelec-tric scanner.

In the present communication we proposean alternative and more straightforward time-multiplexed technique for sampling of the wavefrontfor registration of both the amplitude and the phaseof the optical field. The technique operates withoutany reference beam and without any imaging lensor a microscope objective, providing micrometer-scale resolution, which is comparable to previouslyreported techniques [5–7,12,14]. Digital Micromir-

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ror Device (DMD), initially developed by Texas In-struments for image projection, have been used asimage plane sampling device in spectrometry andlaser beam characterization [15, 16]. In this com-munication we describe the use of DMD for sam-pling the optical field in the pupil plane by seriesof Young’s interferometers, dynamically formed bythe micromirrors of a DMD.

Figure 1: Coherent imaging setup based on fieldsampling with the use of DMD.

Figure 1 explains the principle of registration ofthe complex amplitude for the case of a transmis-sive object, however the same principle can be real-ized for reflective and scattering objects. The lightbeam emitted by laser and scattered by the object,forms a speckle pattern on the surface of DMD. TheDMD device samples the field by turning on onlytwo (reference and sample) micromirrors, to form atwo-point interferometer. The fringe pattern is reg-istered in the focal Fourier plane of the collectivelens. The phase of the fringe pattern, with respectto the origin, is equal to the phase difference be-tween the sample and reference micromirrors. Thephase of the intensity pattern can be easily calcu-lated by making a Fourier transform of the image,and then by calculating the phase in one of sidelobemaxima. The complete sequence, resulting in theregistration of the complex amplitude of the fieldin all points corresponding to micromirrors with in-dices (i, j) and coordinates x = iδ, y = jδ, whereδ is the pitch of the DMD, is given below:

• Chose reference micromirror, by scanning theintensity profile. To ensure high visibility of theinterference pattern, the reference micromirrorshould be placed in an area with high intensity,for instance in a bright speckle. The referencemicromirror stays in the “on” position duringthe whole registration process for all pixels inthe aperture.

• Turn on the sample micromirror with coordi-nates (x, y) and register the interference pat-tern in the focal plane of the collective lens.

Example interferograms obtained with differ-ent micromirror pairs are shown in Fig. 2.

• Process the fringe pattern, to find the phaseϕr(x, y) corresponding to the fundamental har-monic in the registered fringe pattern. Thiscan be done by Fourier transforming the im-age and calculating the phase in one of twosidelobes. Register the intensity in the samplepoint Ir(x, y) by calculating the total intensityin the registered fringe pattern.

• Go to the next sample micromirror, and pro-ceed until field intensity and phase are regis-tered for all points of interest.

Since all micromirrors, except the two that are usedfor interferometry, are in the “off” state, the re-flected beam can be used for conventional imagingduring the whole registration procedure.

Figure 2: Interferograms obtained experimentallyfor different positions of the sampling micromirror.

The systematic phase errors are caused by theaberrations and misalignments in the path of theregistration system, by the geometry of DMD, bythe image noise and the limited size of the im-aged area in the Fourier plane. The major sys-tematic error is caused by the grating-like geom-etry of the DMD. The “on” pixels reflect incom-ing light at the angle of φ = 24◦ in the sagittal(horizontal) plane, introducing path difference of∆ϕ = x sin(φ) ≈ 0.407x where x is the pixel co-ordinate in the sagittal plane. To enable a correctregistration of the complex amplitude, the coher-ence length of the source Lc should satisfy the con-dition Lc � 0.407Ax, where Ax is the full size ofthe registered field in the sagittal plane.

To account for all systematic aberrations, wecalibrated the system by measuring its responseϕc(x, y) to a reference plane wave. Aligned stablesystem has to be calibrated only once. The intensityIo and phase ϕo of the object field can be expressed

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through the measured values of integral intensity ofthe interferogram Ir(x, y) and phase ϕr(x, y), andthe phase of the calibration beam ϕc(x, y):

Io(x, y) = Ir(x, y)−minx,y

[Ir(x, y)]

ϕo(x, y) = ϕr(x, y)− ϕc(x, y) (1)

The lateral resolution r = 0.61λ/NA of the sys-tem is defined by its numerical aperture, which canbe defined as NA = A/2Z, where A is the size ofthe virtual aperture, and Z is the distance betweenthe DMD and the object. The numerical apertureof the virtual optical system can not be larger thanthe angle of diffraction on a single micromirror λ/δ,where we assume the micromirror size is equal tothe pitch δ. With the smallest reported pitch ofDMD of 5.4 µm, and the wavelength varying in therange of 0.4 . . . 1.5 µm, we obtain the NA rangeof 0.075 . . . 0.27, with larger NA corresponding to alonger wavelength. In a similar way, the pixel pitchδ of the DMD determines the smallest spatial fre-quency detectable: λ/δ. Evidently, the field of viewis limited by λ/δ as any two points separated by alarger angle would cause no interference.

Figure 3: Virtual optical system for image recon-struction.

Figure 4: The object for imaging experiments: astage micrometer (R1L3S2P) from Thorlabs. Thecentral line (magnified in the inset) was used forexperiments.

After the complex amplitude U(x, y) =√Io(x, y)eiϕo(x,y) is measured for all points in

the pupil, the field distribution in the object plane

Figure 5: Field intensity (top) and phase (bot-tom) registered in different planes with magnifica-tion M = 1, and a total field of view of 3x3 mm.From left to right the image shows fields registeredat the DMD plane, and back propagated to -2.5 cm,-5.0 cm and -20.0 cm from the DMD.

Figure 6: Image reconstructed in the best focusplane of a virtual optical system shown in Fig. 3,with magnification M = −1, with magnificationM = −2, with M = −2 and one wave of astig-matism, and with M = −2 and one wave of coma(left to right).

can be reconstructed by direct back propagationfrom the plane of the DMD to the object plane, byusing, for example, a spectral method:

U(x, y, Z) = Φ−1[Φ[U(x, y, 0)] · e(−iZ

√k2−u2

x−u2y)],

U(x, y, 0) = Φ−1[Φ[U(x, y, Z)] · e(iZ

√k2−u2

x−u2y)],

(2)

where Φ and Φ−1 are forward and inverse Fouriertransforms, Z is the distance from the object to theDMD, ux and uy are the coordinated in the Fourierspace, and k = 2π/λ, where λ is the wavelength. Di-rect application of expressions (2) reconstructs thefield with a magnification of M = 1. The magnifica-tion can be altered by propagation of the coherentfield through a virtual optical system formed by alens with focal length F , as shown in Fig. 3, where

M = Z1/(Z + ∆Z),

F =Z1(Z + ∆Z)

Z1 + Z + ∆Z. (3)

The lens can be modeled as a phase mask ϕl(x, y) =

k x2+y2

F . In the simplest case of M = 1 and ∆Z = 0,we obtain Z1 = Z, and F = Z/2.

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Our experimental system is formed by a 15 mWHe-Ne laser with a beam expander. The laser il-luminates a low-cost DM365 “LightCrafter” DMDwith a pixel pitch of δ = 10.8 µm. The reflectedlight is collected to the Fourier plane by a two-inchF/3 lens with a focal length of 15 cm. The interfer-ograms are registered by a low-cost CMOS camerawith pixel pitch of 5.5 µm in a region of interest of512x512 pixels. After registration of the referencephase, corresponding to undisturbed laser beam, anobject represented by transparent glass slide ( seeFigure 4) with a high-resolution chrome mask wasplaced in the light path at a distance of ∼5 cmfrom the DMD. The field was registered in a gridof 150x150 pixels, with pixel pitch of 2δ = 21.6 µm,forming a virtual square aperture of about 3x3 mm.The numerical aperture of the virtual optical sys-tem equals to NA = A/2Z = 3/100 ∼ 0.03. Thelateral resolution of the optical system is estimatedas r = 0.61λ/NA ∼ 13µm.

Figure 5 shows the field distribution registeredin the plane of DMD, and reconstructed in threedifferent planes, including the plane of best focus,by using back-propagation described by the pair ofexpressions (2). Figure 6 illustrates the result ofpropagation in a virtual optical system shown inFig. 3 with Z = ∆Z = 0.05 m, and magnificationof M = −1 and M = −2. The last two images inFig. 6 illustrate the possibility of virtual aberrationcorrection by applying correcting phase terms tothe experimentally registered complex field. Sincethe optical system is practically free from aberra-tion, addition of aberration terms reduces the imagesharpness, but proves the point that virtual adap-tive correction is possible in such a system. Sucha correction could be necessary, for example, if theobject image is registered through the cover glass,causing spherical aberration.

Reconstruction of a complex field with dimen-sions of NxN requires N2 interferograms to beregistered and processed. Our experimental setupbased on inexpensive components achieved registra-tion rates of up to 7 pixels per second, mostly lim-ited by the video interface of the DMD and thecamera frame rate. Application of a more advancedDMD and camera would allow to increase the reg-istration rate up to at least 4000 pixels per second,resulting in a time requirement of fifteen secondsper frame for pupil sampling of 256x256 pixels.

Further acceleration can be achieved by increas-ing the number of active pixel in a frame from twoto a larger number, thus forming a sparse intensitymask in the pupil. Such a sparsely sampled pupil

Figure 7: Original phase (left) and phase recon-structed from only 50 frames, using Gerchberg-Saxton algorithm.

produces speckle pattern in the image plane, whichcan be used for the reconstruction of the phase inall the “on” pixels at once, by solving the inverseproblem, for instance using the Gerchberg-Saxtonalgorithm [8]. Our preliminary numerical experi-ments demonstrated full reconstruction of the phasein a pupil formed by 304×342 pixels from only 50frames, with each frame randomly sampled with∼2100 pixels in the “on” state. Each frame with2100 “on” pixels generated a single intensity pat-tern, limited to 255 scales of gray, to represent a8-bit camera image. To obtain the common phasereference, one pixel was designated to be commonbetween all 50 frames. The method yielded phasereconstruction, with only 10 iterations used in eachframe. Such a quick convergence is conditioned bythe randomness and sparsity of the intensity carrierin the pupil: in each frame only 2% of the totalpixels are in the “on” state. An example of recon-struction of a low-order aberration introduced into abeam with uniform intensity distribution, is shownin Fig. 7. Although the first computational resultsare reassuring, a number of issues remains to be ad-dressed in the future development: the speed of con-vergence, noise in the reconstruction, optimal choiceof the common pixels etc. Also, in experiment, thepropagation model used in the Gerchberg-Saxtonalgorithm should be calibrated, to take into accountmisalignments and aberrations of the system. Fur-ther development of the algorithm is possible bychoice of a certain pattern for each frame in the se-quence, for instance by using brighter pixels for thefirst frames, to speed up the collection of useful in-formation. This approach would represent a kindof compressive sensing [17], which in our case willbe employed not to the intensity field, but for thereconstruction of the complete complex field in thepupil of a coherent optical system. The work, aimedto further development of this approach, includ-ing its experimental implementation, is currently in

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progress.In conclusion, we have experimentally demon-

strated registration of complex coherent optical fieldby aperture sampling using digital micromirror de-vice. Unlike other approaches to digital holography,our method does not require any external referencebeam, resulting in a very simple and robust registra-tion setup. Based on this approach, a lensless coher-ent microscope has been realized, in which the com-plex amplitude has been experimentally registeredinside a 3 mm aperture in a grid with 150x150 sam-pling points. The experimentally registered fieldhave been propagated in a virtual optical system todemonstrate virtual imaging, digital focusing, andadaptive correction.

The work of M. Verhaegen, G. Vdovin, P. Pozziand O. Soloviev is supported by the European Re-search Council, Advanced Grant Agreement No.339681. The work of O. Soloviev and G. Vdovinis partly supported by the program “5 in 100” ofthe Russian Ministry of Education. The work ofHai Gong is sponsored by the China ScholarshipCouncil.

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