Image reconstruction using a photonic crystal based flat lens operating at 155μm

Image reconstruction using a photonic crystal basedflat lens operating at 1:55 μm

Maxence Hofman, Didier Lippens, and Olivier Vanbésien*Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN), UMR CNRS 8520,

Université des Sciences et Technologies de Lille, Avenue Poincaré, BP 60069 59655 Villeneuve d’Ascq Cedex France

*Corresponding author: [email protected]‑lille1.fr

Received 23 June 2010; revised 1 September 2010; accepted 25 September 2010;posted 29 September 2010 (Doc. ID 130535); published 15 October 2010

We used an optimized photonic crystal based flat lens for target detection and image reconstruction ofmicrometer sized objects for an operating wavelength of 1:55 μm. Using numerical retrieval proceduresinspired from tomography, the ability to detect subwavelength sized targets and to imagemetallic objectsof complex shapes is shown. The relation between the reconstructed image quality and lens resolution isinvestigated. © 2010 Optical Society of AmericaOCIS codes: 220.2740, 230.5298, 080.2710, 170.6960.

1. Introduction

Since the pioneering works of Veselago [1] andPendry [2] on negative refraction, many papers havebeen devoted to the search of a flat “superlens” ex-ploiting a n ¼ −1 propagating medium based eitheron metamaterials or photonic crystals (PCs) [3–10].Presently, depending on the wavelength of operation,subwavelength or near λ=2 resolutions in focusingexperiments have been theoretically predicted or ex-perimentally estimated using various prototypes.The most promising results in terms of subwave-length resolution (around λ=10) have been obtainedusing metamaterials at microwaves [3]. In thisfrequency domain, the approach used is a simulta-neous engineering of permittivity and permeabilityto fulfill the stringent condition ε ¼ μ ¼ −1 (n ¼ −1,z ¼ 1), supposed to be the sole that allows superreso-lution. Limitations in these cases come from the in-trinsic constituting material losses included in theDrude or Lorentz laws to characterize the artificialmaterial.

At optical frequencies, the situation appears morecomplex since parameter engineering, still theoreti-cally possible, becomes trickier in terms of nanofab-

rication because d ≪ λ, where d is the characteristicpatterning scale, is hard to realize. This explains whyone turns toward PCs, d∼ λ=3, in which negativerefraction is also possible. For such periodic struc-tures, refraction index engineering is well mastered[11,12]. However, impedance matching remains anopen question [13]. This means that the ideal super-focusing condition has not been yet reached in PCs.Despite this fact, promising results have been ob-tained on various prototypes [14–16]. In experi-ments, unknowns on the nature of the source ofoptical wave, on the refractive index or surface impe-dance values, or on the device isotropy often remain,which can lead to misinterpretations of measuredimages. If one considers also the theoretical debateabout the ability of PCs to amplify evanescent wavesin the negative refraction regime, it leads to a rathercontrasted situation about the ability of PCs to over-come the limitations of classical optics [17–23].

In the framework of two-dimensional (2D 1=2) PCsdesigned to operate with an index of refraction equalto −1, near λ=2 resolutions have been theoreticallypredicted (down to 0:4λ for impedance matchedlenses or defect-assisted lenses) and experimentallymeasured (close to 0:8λ by scanning near-field opticalmicroscopy analysis) at optical frequencies (λ ¼1:55 μm) [14]. To go beyond such “simple” focusingdemonstrations, the aim of this paper is to test the

0003-6935/10/305806-08$15.00/0© 2010 Optical Society of America

5806 APPLIED OPTICS / Vol. 49, No. 30 / 20 October 2010

ability of a PC based flat lens to be used for targetdetection and imaging. Up to now, only a few at-tempts have been numerically explored at micro-waves using n ¼ −1 flat lenses characterized byconstituting homogeneous parameters ε and μ forthat purpose [24,25].

Here, a precise full design of the lens based on pre-vious works of our group [14,26], will be consideredfor finite-difference time-domain (FDTD) calcula-tions in order to assess precisely the flat lens perfor-mances for detection and image reconstruction. Tothe best of our knowledge, only PCs allow the designof a flat impedance matched focusing lens withoutoptical axis at near-IR or mid-IR. Using numericaltechniques inspired from tomography [27], we takeadvantage of such a lens to reconstruct a 2D imageof an object located behind the lens.

The paper is organized as follows: in Section 2, theoptimized geometry of the n ¼ −1 PC based flat lenswill be given. In Section 3, the numerical approach tocreate the image of the object is presented. Section 4will illustrate the target detection and image recon-struction for various object shapes, and potential per-formances will be estimated. Section 5 addresses thecomplex relation between the lens resolution andcharacteristic dimensions of the reconstructed im-age. Section 6 will contain concluding remarks andprospects.

2. Optimized n ¼ −1 Photonic Crystal Based Flat Lens

A 2D semiconducting PC based flat lens designed tooperate in the n ¼ −1 negative refraction regimeat 1:55 μm is considered. Details of design andfabrication can be found elsewhere [14,28]. In brief,it consists of a 2D airhole array patterned in a semi-conductor InP based confining heterostructure. For2D FDTD simulations, an effective index neff corre-sponding to the fundamental propagating mode inthe vertical direction is used. Figure 1(a) gives thephotonic band structure for the TE and TM modes.The operating point corresponds to the crossing be-tween the light line and the second band for TE po-larization. For this a=λ value of 0.31, one can notethat a TM gap exists. At this point, the band struc-ture is isotropic and due to the slope reversal, an ef-fective optical index of −1 can be defined. Such adevice has experimentally shown the ability to focusa quasi-point source with a resolution estimatedaround 0:8λ. Theoretically, subwavelength resolu-tions down to 0:35λ–0:4λ are expected owing to air-hole and surface engineering [12]. As the majordrawback of such a flat lens is the wave matching be-tween air and the lens, we follow the procedure ofSmigaj et al. [29] by inserting antireflection coatings(ARCs) on both sides of the lens. Huge increase of thetransmission properties has been obtained theoreti-cally and experimentally [26]. Moreover, it is ex-pected that such enhancement of transmission isrobust versus the light incidence, a property of primeimportance for the image reconstruction envisagedhereafter.

Table 1 summarizes the geometric characteristicsto operate at λ ¼ 1:55 μm with a refractive index of−1 under matched conditions. Simulations are car-ried out using a 2D FDTD code. The fine structureof the device is included in the simulation domain,and, hence, no effective parameters in terms of per-mittivity or permeability are used. Figure 1(b) illus-trates the double focusing numerical experiment(schematized in Table 1). Injection is performedusing a 400 nm wide cut waveguide to mimic a qua-si-point source. We obtain a first focus located in themiddle of the lens for a source/lens distance equal tohalf of the lens thickness. A second focus appears be-hind the lens, symmetric with respect to the sourceposition. It is in accordance with the geometric criter-ion given by the Snell–Descartes law generalized tothe negative index of refraction. We have shown in aprevious paper that this configuration, notably for

Fig. 1. (Color online) (a) Band structure of the PC (the crossing ofthe light line and the second TE band at a=λ ¼ 0:31 defines theoperating point) and (b) focusing experiment at 1:55 μm for the13-row lens described in Table 1.

20 October 2010 / Vol. 49, No. 30 / APPLIED OPTICS 5807

injection, was optimal in such a lens configuration toobtain the higher resolution, here estimated at 0:41λ(640 nm) [12].

As expected, the reflection at the input interface islow due to ARCs. The latter has no major influenceon the resolution, whose optimal value remains un-changed with or without ARCs. As shown in Table 1,one can note that d1 and d2, respectively, the source/lens and the lens/image distance are almost equalbut not strictly equal to half the lens thickness aspredicted for a n ¼ −1 lens. This perturbation is

due to the presence of the ARCs at input and out-put, which modifies slightly the wave propagationthrough the lens over a distance, close to two latticeperiods, not negligible compared with the lens thick-ness. Such an effect does not modify the principle ofoperation of the negative refraction based lens butwill be of prime importance to define properly thefocusing planes of the lens for target detection andimage reconstruction.

3. Principle of Target Detection and ImageReconstruction

Figure 2 illustrates the principle of the setup pro-posed to reconstruct the image of an object locatedbehind the lens. We suppose that the object position,given by its barycenter, is known in the real ðx; yÞspace. Here, for simplicity, the target is supposedto be a perfect reflector. Indeed, for this first demon-stration, only the potentiality of the PC based lens fortarget shape reconstruction will be investigated. Theidea is to collect the backscattered optical field alongthe estimated focusing line in front of the lens for var-ious incidences on the object. Ideally, for a lens thick-ness d, the position of the observation line in front ofthe lens, equal to the lens-target distance, is d=2. Inpractice, we use the simulation results d1 and d2 ofthe optimized lens primarily investigated in Section2. By construction, to exploit the n ¼ −1 lens proper-ties, the distance between the output interface of thelens to the barycenter of the target will have to belower than the lens thickness. Also, as a consequenceof the TM gap existence, no polarizer has been in-cluded in the typical setup because such waves cre-ated by the reflection on nonplanar objects will befiltered by the lens before analysis.

Fig. 2. (Color online) Operating numerical setup for target detection and image reconstruction using a n ¼ −1 flat lens.

Table 1. Photonic Crystal, Antireflection Coatings, and Flat LensParameters for Operation at 1:55 μm with a Refractive Index

of −1 Under Matched Conditions

2D PC:Hole diameter 347 nmLattice period 476 nmFilling factor 48%Effective index (neff ) 3.26ARC:Triangle height (h) 329 nmTriangle base (b) 238 nmSemiconductor

thickness in frontof a hole (t)

33 nm

Flat Lens:Number of rows 13Thickness:—Without ARC 5:36 μm—With ARC 6 μmResolution

at 1:55 μm640 nm (0:41λ)

Focusing lengths:—d1 2:24 μm—d2 2:35 μm


The detection and image reconstruction procedureis as follows [30,31]:

i. The lens is illuminated by a plane wave undernormal incidence. This probing incident wave istransmitted through the lens without significantchange (n ¼ −1, impedance matched, normal inci-dence). The backscattered optical field reflected bythe object located behind the lens is then collectedafter experiencing double negative refraction in frontof the lens. For a given lens/target distance (d2), a “fo-cus” line parallel to the lens first interface is defined(d1). At this position, the optical field is measured.

ii. The operation is repeated for various incidenceangles (θ) on the object, each measure is called a “pro-jection” and labeled pθðtÞ. To create the various inci-dences on the object, the target is supposed to rotatearound its center of mass whereas the lens and in-jected plane wave are fixed. This is equivalent to afixed target with a system lens/injection rotatingaround the target, as shown in Fig. 2.

iii. To reconstruct the image of the object, webuild a table, called a sinogram, containing the dif-ferent parallel projections pθðtÞ. A constant step forthe incidence angle has to be used for the reconstruc-tion in the real space of the image f ðx; yÞ deducedfrom the intensity of the backscattered wave. Themathematical link between the real image and thesinogram is called the Radon transform and is de-fined as

pθðtÞ ¼Zþ∞

−∞

Zþ∞

−∞

f ðx; yÞδðx cos θ þ y sin θ − tÞdxdy;

where the line x cos θ þ y sin θ corresponds to the in-tegration path.

iv. As a last step to reconstruct the image of thetarget, a numerical technique commonly used in to-mography called the “filtered backprojection algo-rithm” can be used to apply the inverse Radontransform to the sinogram [32].

The final image quality depends not only on thelens properties but also on the chosen angle discre-tization. Typically, a regular step of 5° is sufficientto handle the finest details of objects investigatedin this work.

It has been checked that the presence of the flatlens is beneficial for image reconstruction in termsof resolution and quality. To get a quantitative com-parison between the various images, we make use ofa 2D linear correlation coefficient (−1 < r < 1). Avalue close to 1 or −1 means high correlation, anda value of 0 corresponds to two independent images.r contains no information about the respective ampli-tudes of various reconstructed images. Let us men-tion that such a tool is helpful to compare resultsobtained with or without a lens, but it is too simpleto characterize completely the reconstructed imagecompared to the initial target. Indeed, the targets

considered in this work are metallic and, conse-quently, the core part of large objects cannot bereached and restored in the image because the oper-ating frequency lies far below the target plasma fre-quency. Hence, no light can penetrate through theprobed samples.

Figures 3(a) and 3(b) give the results obtained for ametallic crossbar with and without a lens. As a guidefor the eye, the expected shape is redrawn on the

Fig. 3. (Color online) Target detection and image reconstructionfor ametallic crossbar at 1:55 μm: (a) reconstructed image with theflat lens, (b) reconstructed images without the flat lens, and (c) op-tical field in front of the lens without reconstruction. (The object tobe imaged is redrawn on the upper side as a guide for the eye.)


upper side and projected on the reconstructed image.The result is very satisfactory for Fig. 3(a), and thecorrelation coefficient r is found to be close to 0.6 (ap-plied to the full image 15 μm × 15 μm). If the lens isremoved, r drops to −0:1. For Fig. 3(b), also withoutthe lens, we determine the optimal position in termsof correlation. A maximum value of 0.25 is foundwhen the observation line is closer to the target thanwith the lens. As expected, without the lens, the re-flected waves are not “captured” and spread at input.On the contrary, with the lens, information concen-trates along the focusing line in front of the lens.As a consequence, along with the correlation im-provement, the reconstructed image intensity (infor-mation not included in r), which takes advantage ofthe double focusing, is increased. This first compar-ison thus validates the use of a PC flat lens. Detailsabout the image will be discussed in the next section.Moreover, the image reconstruction procedure is ne-cessary to retrieve complex shapes. As shown byFig. 3(c), for a fixed lens/object configuration, the per-turbed reflected optical field indicates the presence ofan object but with poor information about its shapeand position.

Let us mention that, as a consequence of the cur-rent methodology based on the target barycenter asthe reference position, only objects located at a dis-tance less than the lens thickness can be detectedto benefit from the double focusing amplification in-duced by negative refraction. In practice, for PCs,this implies distances not greater than a few operat-ing wavelengths.

4. Numerical Results

The feasibility of target detection and image recon-struction has been tested at λ ¼ 1:55 μm using re-flecting objects (metals) of different shapes. A lensthickness of about 5 μm is chosen, and the barycenterof the target is put at about 2:5 μm of the output in-terface. They differ by their symmetry level and bytheir complexity in terms of concavity and convexity.The angular step and the number of lines to consti-tute the sinogram used to form the image depend onthe symmetry degree of the object. Results are givenin Figs. 3 and 4 for a crossbar [Fig. 3(a)], a circle[Figs. 4(a) and 4(b)], a square [Fig. 4(c)], a rectangle[Fig. 4(d)], and a triangle [Fig. 4(e)].

Figure 4(a) corresponds to the case of a circularscatter for a 600 nm diameter. Such a value is closeto the optimal resolution of such a PC based lens(0:4λ). The circular shape is well reconstructed, butone can note a “halo” around the main trace. Thebrightest part of the image fits well the initial objectsize (seen by projection), and a correlation coefficientof 0.82 is found. Such a result can be obtained over alarge diameter range. The consistency between thedimensions of the reconstructed image and the initialtarget will be discussed in the next section. Let usmention that for objects twice larger than the resolu-tion (around 1:3–1:5 μm), the reconstructed imageappears annular [Fig. 4(b)]. Consequently, the corre-

lation coefficient drops to 0.1 for a 2600 nm diameter.This reveals that the information collected in the in-put focus plane is taken from the backward reflectedwave, which has impinged the target conducting sur-face without penetrating it deeply. As we kept for areference the distance between the lens and the bar-ycenter of the object to extract the optical signal, thecollected amplitude decreases around the center asthe diameter of the object increases.

In Fig. 4(c), a square (each side equal to 2 μm) isconsidered. Here again, lateral dimensions are wellreproduced and the initial target is easily recogniz-able. One can observe that in this case of a large ob-ject, the reconstructed image is not completelybright. Only contours are fully apparent with thebrightest regions corresponding to the corners, andwe obtain a correlation factor of about 0.4. This ap-pears to be the direct consequence of the operatingprocedure chosen. Indeed, let us remember that tobuild the different projections, the object is rotatedaround its barycenter. This means that for a 45° ro-tation, the front corner of the target is closer to theback interface of the lens than for the initial position

Fig. 4. (Color online) Target detection and image reconstructionat 1:55 μm for (a) a 600 nm diameter metallic circle, (b) a 2:6 μmdiameter metallic circle, (c) a 2 μm side metallic square, (d) a4 μm× 1 μmmetallic rectangle, and (e) an isosceles triangle (base:2 μm). (The object to be imaged is redrawn on the upper side as aguide for the eye.)


at 0°. On the other side, it has been decided to keepthe observation line fixed whatever the angle inves-tigated. As a consequence, because most of the inci-dent wave reflection of the object occurs in front ofthe supposed focusing region of the lens definedfor θ ¼ 0°, the information measured at the entrancedoes not correspond exactly to the focus of the back-scattered wave. Such an effect is amplified when thetarget becomes asymmetric. Figure 4(d) illustratesthe case of a rectangle (1 μm× 4 μm) with an equiva-lent correlation coefficient of 0.37. The 1 μm thick-ness is well reproduced, whereas the intensity ismodulated along the 4 μm side. To go further, letus consider the case of an isosceles triangle (base:2 μm), given in Fig. 4(e). The form is again easily re-cognizable even if the correlation factor is quite smallaround 0.16. The limitation comes here from thelarge acute angles to be imaged.

Figure 3(a) illustrates what can be obtained with amore complex shape, such as a crossbar. If the mostangular region appears smoothed, one can recognizeclearly the four arms of the cross showing the abilityto handle quite irregular objects. However, by consid-ering the examples of Figs. 3(a), 4(d), and 4(e), abruptedges appear quite difficult to reconstruct preciselyprobably limited by the lens resolution. It can beshown that the resolution of angular measurementschosen to construct the sinogram influences the im-age quality only at the second order. Values between5° and 10° are sufficient to image objects with at leasta 90° rotational symmetry. This shows clearly the im-portance to reach large subwavelength resolution forsuch flat lenses based on PCs or metamaterials forintegrated nanophotonics. The smaller the resolu-tion is, the higher the ability is to image short rangespatial abrupt corners (obtuse or acute).

5. Discussion

To go further in the analysis, let us reconsider thecircular target of Figs. 4(a) and 4(b) by changingits diameter. What is the limit of detection? Whatprecision can be reached for the reconstructed im-age? To this aim, Fig. 5(a) gives the projectionp0ðtÞ used under normal incidence for various targetdiameters. Let us recall that the circular symmetryimplies that only one recording at θ ¼ 0° contains allthe information needed to build the sinogram and toreconstruct the image. The estimated size of the re-constructed image is given by the full width at halfmaximum (FWHM) of the detected peak. The detec-tion limit will be reached when this amplitude foundfor the FWHM becomes smaller than the mean valueof the optical signal along the observation line. Forthe case of annular images (two peaks along the ob-servation line), the FWHM is defined from the leftside of the first peak to the right side of the secondpeak (in those cases, images are necessary to avoidmisinterpretation).

One can observe that the detection limit is verysmall compared to the wavelength and can reach val-ues around λ=30 (∼50 nm). However, in this case the

maximum amplitude of the measured optical wave islow compared to the background values. Moreover,the FWHM is much greater than the target sizeand appears slightly bigger than the estimated opti-mal resolution of the lens. Figure 5(b) shows such alow limit results undoubtedly from the presence ofthe focusing lens. Without the lens, no valuable infor-mation can be extracted from the recorded signal.

As the target diameter increases, the maximumamplitude of the optical signal follows the sametrend and a good agreement between the FWHM andthe target size is found when the latter becomes closeto the optimal lens resolution, here estimated around600 nm (∼0:4λ). Such a precision is kept up to valuesclose to 3000 nm [∼2λ; see also Figs. 4(a) and 4(b)].For higher diameters, the proximity between the tar-get and the output interface of the object tends to per-turb the reflected optical signal. The precise imagingof a larger object requires a thicker lens in order topreserve the operatingmode of double focusing basedon negative refraction. Indeed, increasing the lensthickness d implies by scaling a joint increase of dis-tances d1 and d2, which always remain by principleclose to d=2. Another approach can be developed toimage larger objects by considering fixed the dis-tance between the output interface of the lens and

Fig. 5. (Color online) (a) Optical signals extracted in the inputfocusing plane for three circular target diameters: 50, 600, and2600 nm. (b) Comparison between the optical signal recorded withand without the lens in presence of a 50 nm circular target.


the target surface instead of its barycenter. Such anextension of our method goes beyond the scope of thispaper dedicated to the detection and imaging ofsmall objects.

For targets with more complex shapes, the reason-ing is similar. If the total size of the object is smallerthan the resolution, only the detecting ability is pre-served with the reconstruction of a circular image.The real shape becomes detectable as soon as eachcharacteristic dimension of the object goes beyondthe lens resolution. Even if sharp corners (for bothobtuse and acute angles) appear quite blurred onthe reconstructed images, shape recognition is possi-ble as long as the ratios between the smallest andthe biggest characteristic dimensions of the objectremain reasonable.

6. Conclusion

The potential of the PC based flat lens for target de-tection and image reconstruction of micrometer sizedobjects has been proposed and evaluated on conduct-ing targets for a wavelength operation at 1:55 μm.Target detection can be largely subwavelength (downto λ=30), whereas image reconstruction depends onthe lens resolution (here around 0:4λ in the best the-oretical case). Inspired from tomography experi-ments, it is believed that such a setup representsa novel approach and a first step toward a micro-meter probe for integrated nanophotonics with theability to detect and image objects with complex geo-metries at the circuit scale. If shape recognition is de-monstrated, the sensitivity of the procedure to theintrinsic nature of the object to be imaged has tobe further investigated.

This work is funded by the Agence Nationale de laRecherche under contract FANI.

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Image reconstruction using a photonic crystal based flat lens operating at 155μm

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