Simultaneous dual-wavelength reflection digitalholography applied to the study of the
porous coal samples
Alexander Khmaladze,1,* Alejandro Restrepo-Martínez,2 Myung Kim,1
Roman Castañeda,3 and Astrid Blandón4
1Department of Physics, University of South Florida, Tampa, Florida 33620, USA2Computer Science School, Facultad de Minas, Universidad Nacional de Colombia, Sede Medellin, AA 3840, Medellin, Colombia
3Physics School, Facultad de Ciencias, Universidad Nacional de Colombia, Sede Medellin, AA. 3840, Medellin, Colombia4Material Science School, Universidad Nacional de Colombia, Sede Medellín, AA 3840, Medellin, Colombia
In conventional holography, the superposition of thewave scattered from the object and the referencewaveresults in an interference pattern, which is recordedon a photographic plate. Then, the hologram is devel-oped throughphotochemical processes [1]. The illumi-nation of the hologram with the reference waveresults in diffraction and propagation of light in sucha way that two images of the original optical field arereproduced, which retain the information of not justamplitude, but also the phase of the original opticalfield.One of them is the orthoscopic (i.e., exact) replicaof the (object) field, while the other is its pseudoscopic(i.e., phase inverted) version.The conventional process of holographic recording
onphotographic plates is rather complicated and time
consuming, which makes real-time imaging difficult.In the past decade, the emphasis has been shifting todigital holography [2]. In this case, the hologram is re-cordedby ahigh-resolutionCCDarray [3–5]. As in thecase of the ordinary holography, the hologram con-tains the information not just of the amplitude distri-butionof light,butalsoof itsphase.After thehologramrecording, the extraction of the amplitude and phasecan be accomplished by numerically propagating thereference wave. The complete and accurate descrip-tion of the propagation of the optical field by diffrac-tion theory allows numerical reconstruction of animage as an array of complex numbers, which repre-sent the amplitude and the phase of the optical field[6]. In addition to the ability of fast image acquisitionand the retrieval of both quantitative amplitude andphase information, digital holography offers the ver-satility of various image processing techniques thatcan be applied to the complex field, which may notbe feasible in real space holography. A number of
differentmethods have been considered for numericalreconstruction includingFresnel transform,Huygensconvolution, and angular spectrum [7–9].Application of digital holography in microscopy is
especially important, because of the extremely nar-row depth of focus of high-magnification systems[10,11]. Microscopic imaging by digital holographyhas been applied to image microstructures and biolo-gical systems [12–14]. Numerical focusing of holo-graphic images can be accomplished from a singleexposedhologram.Direct accessibility to thephase in-formation can be used for numerical correction of dif-ferent aberrations of the optical system, such as fieldcurvature and anamorphism [15]. Phase-contrasttechniques convert the phase changes suffered bythe light wave, while passing through or reflectingfrom objects, into observable intensity variations.Over the years, a number of techniques have been de-veloped to qualitatively perform this conversion. Theexamples include Zernike phase-contrast (ZPC) mi-croscopy and differential interference contrast (DIC)microscopy. InZPCmicroscopy, aphaseplate andspa-tial filtering converts the phase into intensity modu-lation. DIC microscopy, also known as Nomarskiinterference contrast (NIC), uses interferometry oftwo polarized light beams that take slightly differentpaths through the sample. The length of the opticalpaths differ from each other and the beams interferewhen they are recombined, which gives the appear-ance of three-dimensional imaging.ZPC and DIC phase-contrast microscopy techni-
ques cannot be easily used to quantify the phasechange since the conversion of phase to intensitymodulation is nonlinear. Since the phase change in-dicates the change in the optical path length, it canthen be converted to physical thickness, providingthe sample’s height information. Thus, the directaccess to the quantitative phase information makesdigital holographic microscopy (DHM) a true three-dimensional (3D) imaging technique.However, the phase imaging of objects whose opti-
cal thickness variation is greater than thewavelengthof light becomes ambiguous and gives rise to phasewrapping. When the total phase change exceeds 2π,the phase wraps and the image suffers a discontinu-ity.We have previously introduced a dual-wavelengthphase-imaging technique that removes the 2π discon-tinuities by generating two phase images (calledphase maps) using two different wavelengths [16].The usual software algorithmic approaches to phaseunwrapping often are computationally intensive andcannot handle certain complex phase topologies. Thismethod, on the other hand, removes the discontinu-ities by simply comparing the two phasemaps. It thenreduces the phase noise by comparing the resultingdual-wavelength phasemap to one of the original sin-gle-wavelength phase maps. The dual-wavelengthmethod allows imaging to be performed faster, withthe only time constriction being the speed at whichthe two Fourier transforms for both wavelengthsare performed. In the case of a complicated phase
topology, such as porous material, the softwareunwrapping algorithm can mistakenly identify lowintensity areas as multiple phase steps, producingnonexisting height features. On the other hand, indual-wavelength phase imaging, all the observedheight features are real.
Activated coals, as well as coals treated withpyrolysis, are highly porous materials. This is thereason these coals have a high capacity for absorp-tion, which makes them very important in processessuch as purification and filtering. These processes of-ten depend on the size and morphology of the pores.Normally, porosity is evaluated using chemical meth-ods, but these techniques can be rather complex andtime consuming. Optical microscopy and digital ima-ging analysis have been used previously [17] to inves-tigate coal samples, but these methods have theirown limitations in terms of the minimum pore sizethat can be observed. As it was mentioned earlier,ordinary microscopic techniques do not provide the3D depth information about the sample. The porosityevaluation using atomic force microscopy (AFM) failsas the pores are too deep for the tip. Such limitationsin analyzing the pores of coal samples can be over-come by reflection DHM as shown in this work.
2. Experimental Setup
Figure 1 shows the experimental setup, based on twooverlapping Michelson interferometers that enableus to fine-tune the location of the first-order compo-nents associated with each wavelength in the Fourierspace. Its configuration is similar to the setups basedon the modified Mach–Zehnder [18–22] and on theMichelson interferometers [23–25], as typical for
Fig. 1. Multiwavelength digital holography setup. The lateralmagnification of all microscope objectives (OBJ) is 20×. The focallengths of lenses L21 and L22 are 17:5 cm and 10 cm, respectively.The ND filters and polarizers P1 and P2 are used to control theintensity of the laser beams. Pinholes A are used to select onlythe central part of the Gaussian beam. Lenses L11, L12, L21,and L22 and objectives OBJ1, OBJ21, and OBJ22 ensure anappropriate collimation of the waves (i.e., the beam waist is keptat “infinity”).
reflection DHM. He–Ne (λ1 ¼ 633nm) and diode-pumped solid-state (λ2 ¼ 532nm) lasers were usedas light sources. Neutral density filters (ND) controlthe intensity of the laser beams. 20× microscope ob-jectives OBJ11 and OBJ12, together with pinholes Aand the collimator lenses L11=L12, produce uniformplane waves, whose intensity is further controlled bypolarizing filters P1 and P2. Beam splitters BS1 andBS2 divide the beams into the reference and the ob-ject waves, which are reflected by the reference mir-rors and the object. Thereafter, the beam splittersdirect the waves toward the CCD camera.Lenses L21 and L22 and 20× microscope objectives
OBJ21 and OBJ22 again collimate the referencewaves. Two separate reference arms are used tomatch the object path lengths for each object wave.The interference filter, introduced into the referencearm of the diode-pumped solid-state laser, allows onlythis wavelength to pass and blocks the other laser.The 20× microscope objective OBJ1 focuses a magni-fied image of the sample onto the sensor of the CCDcamera, where the interference pattern between thereflected reference and object waves is recorded.An angle between the object wave and each of the
reference waves can be introduced by slightly tiltingthe reference mirrors. Furthermore, by tilting thetwo reference beams orthogonally to each other, wecan precisely control the location of each spectralcomponent in Fourier space [Fig. 2(c)]. As a result,the two spectral components can be sufficiently sepa-rated to enable the effective filtering in the Fourierdomain, which in turn allows for the real-timeimaging.
3. Multiwavelength Phase-Imaging Digital Holography
A. Angular Spectrum Method with the CurvatureCorrection
The angular spectrum algorithm is used for recon-structing the phase and the amplitude informationof the optical field from the recorded hologram. Ithas a number of advantages over the more commonlyused Fresnel transform or Huygens convolutionmethods. The angular spectrum A0ðkx; ky; 0Þ of an ar-bitrary field with intensity distributionE0ðx; y; z ¼ 0Þcan be obtained using the Fourier transform
A0ðkx; ky; 0Þ ¼Z Z
E0ðx; y; 0Þ exp½�iðkxxþ kyyÞ�dxdy;ð1Þ
where kx and ky are spatial frequencies correspond-ing to x and y, respectively. In the case of digital ho-lography, E0ðx; y; z ¼ 0Þ is the intensity distributionrecorded by the CCD camera, which constitutesthe digital hologram, and Eq. (1) is numerically eval-uated. So, the angular spectrum A0ðkx; ky; 0Þ consistsof a zero order and a pair of first-order components,the latter corresponding to the twin holographicimages.
Figure 2(a) displays the image plane digital holo-gram of a United States Air Force (USAF) resolutiontarget, recorded using the experimental setup inFig. 1. The target, which originally was a clear pieceof glass with a chromium pattern on it, was uni-formly covered with a 100nm layer of aluminumto make it entirely reflective. As a result, the patternbecame almost invisible to a naked eye, but the fea-tures were elevated by 100nm with respect to thebackground. The interference patterns of the holo-gram, due to the two wavelengths, can be seen inFig. 2(b), and Fig. 2(c) shows the Fourier spectrumof the hologram, in which the two pairs of first-ordercomponents, corresponding to the two wavelengths,are clearly visible.
The angular spectrum can then be propagated inspace along the z axis as follows:
Aðkx; ky; zÞ ¼ A0ðkx; ky; 0Þ exp½ikzz�; ð2Þ
where exp½ikzz� is the complex transfer function ofthe propagation, with kz ¼
k ¼ 2π=λ. At this stage, the Fourier-domain filteringcan be applied to block the unwanted spectral termsand to select each of the first-order components. Itcan be performed with a numerical band-pass filterif the off-axis angle of the reference beam is properlyadjusted. Finally, a complex wave field is recon-structed by simply performing the inverse Fouriertransform, i.e., numerically evaluating the integral:
Eðx; y; zÞ ¼Z Z
Aðkx; ky; zÞ exp½iðkxxþ kyyÞ�dkxdky:ð3Þ
A significant computational advantage of the an-gular spectrummethod is that once the angular spec-trum at z ¼ 0 is known, only one additional Fouriertransform should be applied for reconstructing theobject wave at any plane z. Furthermore, there isno requirement for z to be larger than a certain mini-mum value, as in the case of the Fresnel integral.
In spite of the collimating configuration of the ex-perimental setup (Fig. 1), some wavefront curvatureremains in each wave. Moreover, because these cur-vatures are not equal, a curvature mismatch appears
Fig. 2. Two-wavelength DHM of a USAF resolution target: (a) di-gital hologram, (b) small section of the hologram in (a) showing twofringe patterns due to the recording wavelengths (the rectangularshadows are two bars of the target image) and (c) Fourier spectrumof the hologram in (a); the encircled sections are the first-ordercomponents for the red and the green wavelengths, respectively.
in the final phase image as shown in Fig. 3(a). Tounderstand this, consider the complex field capturedby a CCD matrix [see Fig. 3(c)]. The phase mismatchcan be compensated for numerically, by multiplyingthe original “flat” field E0ðx; y; z ¼ 0Þ by the phasefactor exp½iϕ�, where ϕ ¼ kd is the phase differencebetween points a and o and d is the optical pathdifference. From geometry, ðRþ dÞ2 ¼ R2 þ r2, orRþ d ¼ �
which is the exact expression for the curvature cor-rected field. The value of R can be experimentallydetermined for a given setup. The sign in Eq. (4) de-pends on whether the image of the flat area is convexor concave. By adjusting this sign and the magnitudeof R in the optical setup, the residual mismatch inthe optical wavefront curvature is properly compen-sated [Fig. 3(b)].
B. Phase and Height Maps
The object surface is described by height map hðx; yÞ,which is determined from phase map ϕðx; yÞ of theholographic reconstruction at a given wavelength by
hðx; yÞ ¼ λ4π ϕðx; yÞ: ð5Þ
Figure 4(a) shows the phase map of the alumi-num-covered USAF resolution target. The step size
is approximately 2:2 rad: According to Eq. (5), it isproportional to the target height map, whose profilecorresponding to the line over the phase map inFig. 4(a) is sketched. The step height is about100nm. This result is confirmed by the AFM scanshown in Fig. 4(b).
C. Multiwavelength Phase Imaging
As stated before, the phase images of objects with var-iations in optical thickness greater than the wave-length are ambiguous and result in phasewrapping. Consequently, the phase map exhibits dis-continuities at the positions where the total phasechange exceeds 2π. However, if the simultaneousdual-wavelength phase imaging is performed, the dis-continuities of the twomaps will occur at different po-sitions, since the two wavelengths are different. Itallows unwrapping the phase by comparing the twomaps, as stated below. In this way, the 2π jumps areremovedand then thephaseambiguities are resolved.
Figure 5 shows the phase images of the USAF re-solution target imaged at an angle. The images pro-duced with single wavelengths exhibit multiplephase steps [Fig. 5(a) and 5(b)]. Phase maps ϕ1andϕ2, derived from each wavelength are subtracted,so that ϕ12 ¼ ϕ1 � ϕ2 is obtained. Adding 2π wher-ever ϕ12 < 0 yields a new phase map, practically freeof discontinuities. It is equivalent to a phase map cre-ated by a single synthetic “beat” wavelength
Λ12 ¼λ1 λ2=���λ1 − λ2
���: ð6Þ
For wavelengths λ1 ¼ 633nm and λ2 ¼ 532nm, thebeatwavelengthisΛ12 ¼ 3334nm[Figs.5(c)and5(d)].Notice thatwhile the phase images produced by a sin-glewavelengthexhibitmultiplediscontinuities, in thefinal synthetic wavelength map the discontinuitiesare removed. In fact, here the synthetic wavelengthis such that the range of the dual-wavelength phasemap is barely enough to resolve the discontinuities[some even remain on the left and right of Fig. 5(c)].
Fig. 3. Reconstructed phase image of the USAF resolution target(a) without curvature correction and (b) with curvature correction(the vertical scale is in nanometers); (c) the diagram illustratingthe curvature correction procedure. R is the wave’s radius of cur-vature, which can be determined experimentally for a given setup.r ¼
pis the distance between the center of the CCDmatrix
(point o) and an arbitrary point a. x and y are the coordinates of a.
Fig. 4. (a) Phase map and height profile for λ ¼ 633nm. The pro-file is taken along the line over the phase map. (b) AFM image andheight profile that confirm the results in (a).
D. Application of the Phase Noise Reduction Algorithm
The drawback of the dual-wavelength method is theamplification of phase noise by the same factor as therange. Furthermore, the two phase maps differ intheir noise distributions, so that the final dual-wavelength phase map can remain quite noisy evenif the noise in the single-wavelength phase maps islow. However, one can use this dual-wavelength“coarse” map as a guide, together with one of the ori-ginal phase maps (ϕ1 or ϕ2), to produce the low noise“fine” phase map. The method (detailed in Ref. [16])involves dividing the height of the coarse map intothe integer number of one of the original wave-lengths, say λ1. Then, the wavelength high segmentsfrom phase map ϕ1 are pasted into this coarse map,which achieves the desired effect of reduced noise to-gether with extended range. In practice, the areasnear the boundaries of the wavelength intervalsare somewhat problematic. There, the noise presentin single-wavelength map ϕ1 causes the height tochange erratically by one wavelength. In order topartially solve this problem, one can compare thismap to the coarse map and, if the difference is morethan λ1=2, add or subtract λ1 depending on the sizeof the difference. However, if the noise is excessive,the last step results in the shift of small portionsof the final image by λ1 from its true position. Sincethe height of such a shift is always λ1, it can be par-tially fixed by software by looking for the steps of thisheight and shifting them up (or down) by λ1.Indeed, the high noise level makes it very difficult
to measure the height of the object’s individual fea-tures from the dual-wavelength phasemap, as shownby the height profile in Fig. 6(a) obtained from themap in Fig. 5(c), despite the fact that the overallshape of the object is still preserved by the dual-phase map. In contrast, the height profile of the fine
map, shown in Fig. 6(b), again yields the step heightof 100nm.
To numerically estimate the noise levels in the sys-tem, the height profiles of the flat area for a singlewavelength, a coarse map, and a fine map (Fig. 7)were taken and their rms noise was measured. Whilethe rms noise in the coarse map is substantial (on theorder of 54nm), the noise for the fine map is almostequal to the single-wavelength phase map (on the or-der of 8:5nm and 6:5nm, respectively).
4. Imaging of Porous Coal Samples
The technique outlined here has been successfullyapplied to the 3D imaging of coal samples. Thesamples were acquired from the Coal Group at theNational University of Colombia, Campus Medellin.These samples were treated by pyrolysis and pre-pared in a mixture with epoxy resin; then they wereground and polished with aluminum oxide abrasives(1, 0.5, and 0:03 μm grain size). The process ensuredthat the samples are reflective and firm enough to beimaged.
Fig. 6. Height profiles of (a) coarse and (b) fine phase maps.(c) Final fine map and (d) 3D rendering of (c). The image sizesare 174 × 174 μm2.
Fig. 7. Line intensity profiles of a flat area for the coarse, fine,and the single-wavelength phase maps, respectively.
Fig. 5. Phase maps for (a) λ1 ¼ 532nm and (b) λ2 ¼ 633nm.(c) Synthetic dual-phase map with beat wavelengthΛ12 ¼3334nm and (d) its 3D rendering (the images are 174 ×174 μm2 and the vertical scale for (a)–(c) is in radians).
In Fig. 8, a boundary between the porous coal and theresin is shown. The amplitude image [Fig. 8(a)] issimilar to what a regular microscope would display.The area corresponding to the resin is at the lowerright corner. It is apparent that the boundary isbarely visible in the amplitude image, as there isno significant difference between the reflection fromthe coal and resin.Figures 8(b) and 8(c) show the single-wavelength
phase images, which display multiple discontinuities
that were removed in the dual-wavelength coarsemap [Fig. 8(d)]. The phase noise is significantly re-duced in the fine map [Fig. 8(e)], as appreciated inits 3D rendering [Fig. 8(f)]. The latter image showsthe 3D surface profiles of the coal and the resin sur-faces, which appear to “bend” towards each other atthe boundary between the two surfaces, so theboundary itself is seen very clearly.
B. Comparison between Dual-Wavelength and SoftwarePhase Unwrapping
Figure9showsthe imagesofporous coal samples trea-ted with pyrolysis. Once again, the phase images at asingle wavelength clearly exhibit 2π phase steps[Figs. 9(b) and 9(c)], while the dual-wavelength un-wrappedphasemap inFig. 9(d) showsvery fewdiscon-tinuities. The parts of the images where discontinuityis still present correspond to low reflectivity areas onthesamplehologram,whereno interferencepattern isvisible.Consequently, thephase there isbasically ran-dom noise, which gives rise to multiple 2π phasejumps. These can generally be identified as deeperpores. With this method, pores with a lateral sizeon the order of a micrometer can be identified.
Fig. 8. Images of a porous coal sample: (a) amplitude image; phasemaps reconstructed at (b) λ1 ¼ 0:63 μm and (c) λ2 ¼ 0:53μm; (d) thedual-wavelength coarse phase map, (e) fine map, and (f) its 3D ren-dering. All images are 98 × 98 μm2 and vertical scale (b–e) is inradians.
Fig. 9. Images of a porous coal sample: (a) amplitude image; phasemaps reconstructed at (b) λ1 ¼ 532nm and (c) λ2 ¼ 633nm; (d) 3Drendering of the dual-wavelength phase map; software unwrappedphasemaps reconstructed at (e) λ1 ¼ 633nm and (f) λ2 ¼ 532nm forcomparison. All image sizes are 98 × 98μm2. The vertical scales ofthe phase maps are in radians.
It is worth noticing that unwrapping the single-wavelength phase images using conventionalalgorithms is very problematic, as illustrated inFigs. 9(e) and 9(f). A typical software unwrapping al-gorithm starts at a certain point of an image andmoves along a one-dimensional (1D) path (e.g.,straight line, spiral). If it encounters what looks likeaphasewrap, it shifts themapdownorup. If the imagehas noisy areas [corners in Figs. 9(e) and 9(f)] wherephase oscillates randomly, the software algorithmtakes it as a real feature and createsnonexistent stepsin phase/height profile, which clearly do not corre-spond to the real height profile of the sample.
C. Application of the Fine Map Algorithm
Figure 10 shows the images of an activated porouscoal in a resin sample. The areas on the left andright sides of the image are too dark, but the centralregion is well illuminated. Two vertical scratches dueto the coal-polishing process are clearly visible. Fig-ures 10(a) and 10(b) represent the coarse and the finephase maps, respectively. A noise reduction by a fac-tor of 5 is apparent by comparing the height profilesin Fig. 10(c) and 10(d) from the coarse and the finemaps, respectively. For further comparison, the sin-
gle-wavelength phase map and the same line profileare sketched in Figs. 10(e) and 10(f), respectively.
The “spots” in the images Fig. 10(c) and 10(d), aswell as the spike around 120 μm in Fig. 10(f), resultfrom the high level noise in the single-wavelengthimages. Consequently, some of the wavelength seg-ments are erroneously shifted by a wavelength(see Subsection 3.D). Artifacts like these are rareand they do not prevent us from obtaining an accurate3D picture of the sample. From the depth informationavailable from the fine map [Fig. 10(d)], one can see,for example, that the depth of a small feature (scratchlocated around 75 μm from the left on all the lineprofiles) is about 100nm, which is again consistentwith the single-wavelength phase map [Fig. 10(f)].Obviously, the single wavelength possesses multipleartifacts due to phase wrapping.
5. Conclusion
The dual-wavelength phase-imaging digital hologra-phy technique proved to be a powerful method of 3Dimaging with the 2π ambiguity resolved. Its applica-tion to thedetectionand study of pores in coal sampleshas been demonstrated. Themethod is advantageousin comparison to the software unwrapping ap-proaches as it requires no intensive computation pro-ceduresandcanhandle complexphase topologies.Themethod provides high-resolution, accurate quantita-tive profiles of surfaces and can be an effective toolin studying small and large scale 3D features of manynatural andman-made samples. The use of twowave-lengths together with the fine map algorithm allowsus to increase the maximum height of the featuresthat can be imaged, while keeping the noise low(few nanometers). The same method can also easilybe applied to imaging in transmission rather than re-flection, which can be extremely useful for imaging ofbiological samples and obtaining the density profilesof cells. The simultaneous dual-wavelength setup uti-lized together with the angular spectrum algorithmprovides an easy way to acquire single frame imagesin real-time,which canbeused to study cellmigration,for instance.
The selection of two wavelengths that are closer toeach other increases the axial range, but also in-creases the noise to the levels where the fine map al-gorithm begins to fail. To further increase the axialrange and still keep the final noise levels low, thesame procedure can be applied at three or morewavelengths. Furthermore, it is possible to use a tun-able laser to iteratively increase the range while re-ducing the noise to the desired levels [2]. Theproposed method of curvature correction is simpleenough to easily implement the same experimentwithout the microscope objectives in the referencearms of the Michelson interferometer. This wouldgreatly simplify the optical setup and make the initi-al adjustments of the interferometers much easier.
This work is, in part, supported by Colciencias,DINAIN, DIME, and the Coal Petrography Labora-tory. The authors also acknowledge Richard Everly
Fig. 10. (a) Coarse phase map, (b) fine phase map, and line pro-files for (c) coarse and (d) fine. For comparison, (e) single-wavelength phasemap at λ ¼ 633nm and (f) line profile (the imagesizes are 138 × 138 μm2 and the vertical scales of the phase imagesare in radians).
of the Nanomaterials & Nanomanufacturing Centerat the University of South Florida for his work inmaking the aluminum coating for the USAF resolu-tion target and Joshua Robinson at the Departmentof Physics at the University of South Florida for hishelp in imaging the target using AFM.
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