During the past 30 years a number of methods forintegrating microlenses on optical fibers �MOFs�have been invented and revisited. MOFs have beenmanufactured by various methods such as electric arcmelting,1 laser micromachining,2 and chemical etch-ing.3 Nevertheless, these procedures are time andenergy consuming and imply stringent control of theexperimental parameters.
In 1974 an elegant alternative method was intro-duced by Cohen and Schneider4 who produced MOFsby exposing to UV light a photoresist film that coatedthe top end of a fiber. Although this procedure wasrevised and improved a few years later,5,6 it requiresdifficult fabrication steps, including the backing of
both the photoresist and the microlens. In fact, asurvey of the literature reveals that the concept ofmaking MOFs by photolithography has not been de-veloped widely despite the promising results pre-sented in Ref. 4.
As reported in Ref. 4, a solid film of negative-tonephotoresist undergoes, by invisible radiation, photo-chemical reactions that effect chemical and physicalchanges that can be developed.7 In this paper, wereport a method of free-radical photopolymerization7
on the top end of both single-mode and multimodeoptical fibers. The process is very flexible, simple tocarry out, and uses the visible light guided in thefiber. It permits one to produce, on the fiber end, amicrometer-sized polymer tip that may be viewed asan extension of the fiber core.
The paper is divided into the following four sec-tions: In Section 2, we describe the polymerizableformulation. In Section 3 both the principle of ourapproach and experiments are presented. Also inSection 3 numerous experimental results, obtainedwith both single-mode and multimode optical fibers,are shown and commented on qualitatively. Section4 is devoted to an interpretation of the experimentalresults. We perform a numerical calculation con-sisting of an iterative beam-propagation method�BPM� into which photochemical laws have been in-troduced. The results of these calculations permit
R. Bachelot �[email protected]�, D. Deloeil, and P. Royerare with the Laboratoire de Nanotechnologie et d’InstrumentationOptique, Universite de Technologie de Troyes, 12 rue Marie Curie,B.P. 2060, 10010 Troyes cedex, France. C. Ecoffet and D.-J.Lougnot are with the Departement de Photochimie Generale, EcoleNationale Superieure de Chimie de Mulhouse, Centre National dela Recherche Scientifique, Unite Mixte de Recherche 7525, 3 rueAlfred Werner, F-68093 Mulhouse, France.
Received 4 August 2000; revised manuscript received 27 March2001.
us to understand the process of tip formation and toconfirm the quantitative interpretations proposed atthe beginning of Section 4. It should be pointed outthat the main purpose of the present paper is both topresent our method of fabricating a polymer tip at theextremities of optical guides and to propose, bynumerical simulation, an interpretation of the tipformation �Sections 3 and 4�. Two potential appli-cations are proposed in Section 5, and preliminaryexperimental results concerning these applicationsare presented. In Section 6, we present conclusions.
2. Photopolymerizable Material
The photopolymerizable formulation used was re-cently introduced in previous papers.8–11 It is madeup of three basic components: a sensitizer dye, anamine cosynergist, and a multifunctional acrylatemonomer, pentaerythritol triacrylate, which is usedas received from the supplier and forms the backboneof the polymer network. The cosynergist was meth-yldiethanolamine; eosin Y �2�, 4�, 5�, 7�-tetrabromo-fluorescein disodium salt� was used as the sensitizerdye. This system was developed mainly because ofits high sensitivity in the spectral region from 450 to550 nm �maximum at 530 nm� and, in particular, toargon laser light �514 nm�. In addition, this liquidsystem is very flexible as it makes it possible to mod-ify the components independently to adjust the phys-ical and the chemical properties of the formulation�viscosity, spectral sensitivity, polymerization thresh-old energy, and so on�. The results reported in thispaper were obtained with mixtures containing 0.5% inweight of eosin and 8% in weight of methyldiethano-lamine. The formulation is liquid and can easily bewashed out with a solvent such as methanol.
After the absorption of actinic light by eosin thetriplet state of the dye reacts with the amine to formradicals. Radicals initiate the polymerization of themonomer. Because of the monomer’s multifunction-ality, the polymer quickly develops into a three-dimensional �3-D� network. The sensitivity of thesystem is characterized by a curve that shows thedegree of cross-linking as a function of the receivedenergy Er. This curve, presented in Fig. 1, corre-sponds to the typical behavior of a formulation thatcan be polymerized following a radical process. Onecan note that polymerization starts only when theabsorbed energy is greater than a threshold valueEth. Typically, this threshold energy, when the O2-diffusion process within the formulation is neglected,is �100 mJ�cm2. The inhibition process arises fromthe sensitivity of the initiating radicals to oxygen.7,10
The triplet state of the dye is readily deactivated byoxygen either through a physical process �singlet ox-ygen is then generated� or through a chemical reac-tion �resulting in peroxidation of the excitedprecursor�. Hence, as long as some quencher re-mains present in the formulation, this side reaction ispredominant, and no radical forms. Radical forma-tion becomes the predominant channel of triplet de-cay when the residual oxygen concentration is lessthan approximately 10�5M. In this case, polymer
chains start growing, and the cross-linking reactionproceeds rapidly. Gelation does not occur immedi-ately, and, at the onset of polymerization, the me-chanical properties of the material are not strongenough to stand development by use of rinsing. Me-chanical resistance appears when the energy re-ceived by the system is greater than the gelationenergy. For now it should be noted that the opticalrefractive index of this formulation varies as a func-tion of the degree of cross-linking from 1.48 �0% cross-linking� to 1.52 �100% cross-linking�.
3. Experiments
A. Experimental Procedure
The experimental setup that was used is shown inFig. 2. A drop of the photosensitive liquid formula-tion is deposited at the end of a cleaved optical fiber.The shape �height and radius of curvature� of thedrop is well reproducible and governed by variousparameters, such as the angle of contact of the for-mulation on the fiber cladding, the formulation vis-cosity, and the fiber diameter. With the formulation
Fig. 1. Reticulation rate of the formulation plotted versus theabsorbed energy. Eth is the threshold energy �when it is achievedpolymerization starts�, and n is the refractive index of the formu-lation.
Fig. 2. Experimental arrangement for exposing a drop of pho-topolymerizable formulation that is deposited over the core of anoptical fiber.
that we used, the height of the drop is �30 �m,whereas its radius of curvature is �100 �m.
Photopolymerization occurs on green light’s actingon the formulation emerging from the core of thefiber. This light is coupled from an argon laser �� �514 nm� into the fiber and is guided by the fiber coreto the fiber end, where the formulation is deposited.Typically, the output intensity Io, measured beforedepositing the drop, is 10 �W, and the exposure timet is in the 1–120-s range. It should be noted that,during exposure, yellow fluorescence arising from eo-sin can be observed in the illuminated area.
After curing the illuminated area of the formula-tion has been polymerized. The unreacted liquidformulation is washed out with a few drops of meth-anol.
B. Photopolymerization on a Single-Mode Optical Fiber
1. Characteristics of the Optical FiberFor this experiment, we used a cleaved single-modeoptical fiber. The characteristics of this fiber are
• Cladding diameter, 125 �m.• Core radius, a � 1.5 �m.• Numerical aperture, NA � 0.12.• Refractive index of the core, ncore � 1.54.• Refractive index of the cladding, nclad � 1.53.
From these parameters the waveguide’s V parameterfor this fiber is calculated by12
V �2aNA
�0, (1)
where �0 is the wavelength of the guided light ��0 �514 nm�.
Thus one finds that V � 2.2. This value is inaccord with the condition that V 2.45 is required forsingle-mode propagation within the fiber. At theoutput of such a fiber the magnitude of the electricfield vector W0 varies approximately as a Gaussianfunction of the distance r from the core center
W0�r� � A exp��� rw�2� , (2)
where A represents the magnitude of the optical fieldat the very center of the core and w, the waist of theGaussian beam, can be related to the fiber parame-ters
w ��0
NA. (3)
In our case w � 1.43 �m.
2. Experimental ResultsIn this subsection, we describe only our experimentalobservations. The interpretation of the presentedresults is the subject of Section 4.
Figure 3 presents an electron micrograph of a poly-mer tip realized by our method. For tip fabrication,
the fiber output intensity Io was 10 �W, and theexposure time t was 30 s. Figure 3�a� shows an im-age of the fiber’s top end. It shows a solid polymertip firmly attached to the center of the cleaved fibersurface, just over the core. The tip is 30 �m high,which corresponds to the height of the deposited drop.The tip-base diameter is approximately 4 �m,whereas the extremity is a smooth hemisphere whoseradius of curvature is 1.5-�m � 5% �Fig. 3�b� . Theexperiment was also performed with 670-nm wave-length light. In this case, it was impossible to pro-duce any polymer tip, which demonstrated that theobtained tip was formed by pure free-radical photopo-lymerization that needed eosin as a photoinitiator,thus excluding any photothermal effect in the gener-ation of the polymer element.
Figure 4 and Fig. 6 �below� show the flexibility ofthe procedure, whereas Fig. 5 demonstrates its re-producibility. Figure 4 presents a different set oftips that were obtained with various exposure times�t � 2, 45, 60, 90 s�. These values of t led to tipswhose respective radii of curvature were 0.6 �m �Fig.4�a� , 2.5 �m �Fig. 4�b� , 6.5 �m �Fig. 4�c� , and � �flat
Fig. 3. Electron micrographs of the polymer tip that is formedover the core of a single-mode optical fiber. �a� A 150 �m � 118�m image: the cleaved fiber surface of the polymer tip whose basecoincides with the fiber core. �b� A 5.6 �m � 4.7 �m image show-ing the tip’s extremity.
tip end; Fig. 4�d� . One can note that the longer theexposure time, the greater the radius of curvature ofthe tip extremity. Figure 5 presents the extremitiesof three polymer tips that were fabricated by use ofthe same exposure parameters as for Fig. 4�a�: Itshows the reproducibility of the experiment. InFigs. 3 and 4 it is worth noting two facts:
�i� The micrometer-sized radius of curvature of thetip extremity is much smaller than that of the initialliquid drop ��100 �m�.
�ii� Even for shortest exposure the length of the tipis equal to the drop height ��30 �m in this case�.
Fact 2 is confirmed by Fig. 6, which shows a polymerelement integrated on the cleaved fiber end on whichwe deposited a 2-�m-thick layer of the formulationinstead of a drop. In the case of Fig. 6 a microlenswas integrated at the tip end.
C. Photopolymerization on a MultimodeOptical Fiber
To test the various possibilities of our technique, weperformed the experiment on a multimode opticalfiber. The fiber used had a 4.5-�m core radius sur-rounded by a 125-�m-diameter cladding. Althoughthe fiber has a single mode at � � 1.55 �m, it has four
or five linearly polarized �LP� modes at � � 514 nm.By applying mechanical strains, we selectively ex-cited the fiber’s LP modes and successfully fabricateda 3-D mold of their respective intensity distributions.Two examples are shown in Fig. 7. Figure 7�a�shows the calculated intensity distribution of theLP11 mode of the fiber; Fig. 7�b� shows the corre-sponding polymer element fabricated at the tip end.Figure 7�c� shows the calculated intensity distribu-tion of the LP21 mode of the fiber; Fig. 7�d� shows thecorresponding polymer element fabricated at the tipend. In addition to the interest in characterizing thelight intensity in the vicinity of the opticalwaveguide, we shall see in Section 5 that the obtainedpolymer element can permit the easy selective cou-pling of a specific mode in the fiber.
4. Interpretation of the Tip Formation
A. Preliminary Discussion
The shape of the polymer elements obtained by thislithographic technique was correlated with the char-acteristics of the fiber output beam. This modifica-tion can be evaluated in accord with the responsefunction of the material given in Fig. 1.
As a first approximation, we calculated the re-ceived energy as the product of the time multiplied by
Fig. 4. Electron micrographs of the extremities of four tipped fibers that were fabricated with various exposure times: �a� 2 s, �b� 45 s,�c� 60 s, �d� 90 s.
the intensity within the formulation. The electricfield W0�x, y� at the output of the fiber is given byrelation �2�. If one considers that the formulation isa homogeneous medium with a refractive index of1.48, one can calculate the electric field distributionW�x, y, z� within the drop of formulation. The in-tensity is proportional to the square of the electricfield
I� x, y, z� � W� x, y, z�W*� x, y, z�, (4)
where the asterisk denotes the conjugate form of W,and the absorbed energy Eabs is proportional to thisintensity and to the illumination time t
Eabs � KW� x, y, z�W*� x, y, z�t, (5)
where K is a factor of proportionality that depends onthe formulation. After an exposure of time t theshape of the object formed by polymerization is givenby identification of the absorbed energy and the ge-lation energy.
The parameters that can describe the photopoly-merization kinetics of the formulation are Eth and tth.Eth is the threshold energy below which no polymer-ization occurs in the drop of formulation �see relation�2� :
Eth � KA2tth. (6)
The threshold time of exposure tth is the time whenthe first part of a polymer appears in the drop:
Eabs�Eth �W� x, y, z�W*� x, y, z�
A2
ttth
. (7)
Let us take as a unit of time tth. Consequently, allthe phenomena considered here are studied with thefollowing normalization conventions: Eth � 1, tth �1, and A � 1. For applications these parameters canbe evaluated in an adequate unit system.
Photopolymerization processes are often used inthe fabrication of micro-objects.7,13 The areas ofsuch objects that receive enough energy solidify.Thus the shape of the polymer parts produced by thislithographic technique can be correlated with theoutput-beam characteristics. If one considers thatthe material properties are a function of the absorbedenergy, one can evaluate them as a function of time.In Fig. 8 are represented the isointensity curves forvarious values of the intensity. The value of thetime that is indicated beside each curve correspondsto the theoretical value necessary to reach the thresh-old of the received energy. In other words, thesecurves correspond to the polymerized part that isexpected if the received energy is equal to the initialintensity multiplied by the exposure time t. Thetime unit, as explained above, is the beginning of thepolymerization process at the very center of the out-put fiber.
From Fig. 8, one can note that, when polymeriza-tion occurs at the output of the formulation drop, oneexpects an enlargement of the polymer part. Thisphenomenon was not observed on the polymer part
Fig. 5. Electron micrograph of �a�, �b�, �c� the extremities of threetipped fibers fabricated under the same exposure times as for Fig.4�a�.
Fig. 6. Electron micrograph of a 2-�m-thick microlens fabricatedover the fiber core.
that we obtained experimentally �see, for example,Fig. 3�a� . In fact, it seems that the light is self-guided by the polymer part during its formation. Anextra experiment was performed that showed thisphenomenon more clearly: A single-mode fiber wasdipped into a thick layer of formulation. In this waya long tip �approximately 0.5 mm long� was obtained
�see Fig. 9 as an example�. No enlargement wasobserved. An empirical way to interpret this phe-nomenon is to remember that the refractive index forpentaerythritol acrylate is 1.48 for the monomer and1.52 when the material is polymerized10 �see Fig. 1�.Hence, as soon as the formulation was polymerized, itwould act as an optical waveguide �with the core asthe polymerized area and the cladding as the sur-rounding unpolymerized material�, which would form
Fig. 7. Fabrication of the 3-D polymer molds of the LP modes of a multimode optical fiber. The objects are fabricated just over the fibercore. �a� Theoretical intensity distribution of the LP11 mode, �b� electron micrograph of the fabricated mold of the LP11 mode, �c�theoretical intensity distribution of the LP21 mode, �d� electron micrograph of the fabricated mold of the LP21 mode.
Fig. 8. Numerical simulation of the formation of a polymer tip atthe extremity of a single-mode fiber. The modification of the re-fractive index of the medium during tip formation is not taken intoaccount. t, exposure time �arbitrary units�.
Fig. 9. Electron micrograph of a long-tip fiber that was obtainedby the dipping of the fiber into a thick layer of the formulation.
as the light propagates in this guide, preventing thelight from diverging. Moreover, if it is assumed thatan index gradient exists the divergence effect can beattenuated as the light propagates.
B. Numerical Calculation
To point out the important parameters that governtip formation, we carried out a numerical calculationfor the case of a single-mode fiber. The modificationof the refractive index within the drop of formulationis evaluated as a function of the exposure time. Ageneral flow chart of this method is presented in Fig.10. The definitions of the x and the z axes are pre-sented in Fig. 11. The cleaved output end of thefiber is placed at z � 0, and the light propagates alongthe z axis. The formulation–air interface is at z � 30�m. This value corresponds to the height of thedrop of formulation under the experimental condi-tions. x represents the lateral coordinate perpendic-
ular to the fiber axis. The steps �from 0 to 7� of themethod shown in Fig. 10 are described and discussedbelow.
The light field W�x, z� within the formulation isevaluated by the BPM. This method is particularlysuited to beam propagation within inhomogeneousmaterials. For each iteration, W�x, z� is calculatedby a two-dimensional BPM method that uses bothW�x; z � 0�, as the initial values, and n�x, z�, as theindex distribution �Fig. 10, steps 1 and 2�. The BPMthat was used is classical and very well known. Theprocess of calculation is described in detail in Ref. 14.It consists of successive Fourier transforms and in-verse Fourier transforms for each dz-length section.The influence of the index contrast is represented byan array of lenses that are placed at a distance dzfrom each other. Between each lens a transfer func-tion of the Fourier space describes the light propaga-tion. Each lens imposes, in the direct space, a phasefi�x� that contains the index contrast n�x� at the zplane. As was noted in Ref. 14, the BPM processthat we used is certainly valid if two conditions aresatisfied:
�i� In section dz, the variation of the refractive in-dex in the z direction must be gradual compared withthe wavelength of the light.
�ii� The total distance of propagation must not betoo long �to avoid the accumulation of computationerror�.
In our case these two conditions are satisfied.It is important to remark here that the comparison
between the experimental results and our BPM mod-eling is �can be� only qualitative. Indeed, our simu-lation of the experiment is two dimensional, and thisfact must be kept in mind in our interpretation.
At the output of the optical fiber the electromag-netic field can be considered to be Gaussian and pla-nar; therefore the field W0�x� � W�x; z � 0� is knownand constant. It is taken as the initial value for theBPM calculation �Fig. 10, step 1�. The refractive
Fig. 12. Threshold energy Eth of the reticulation plotted versusthe distance z from the fiber end. The formulation–air interface isat z � 30 �m.
Fig. 10. Principle of the numerical calculation that permits thesimulation of tip formation. See text for details.
Fig. 11. Two-dimensional geometry of the calculation describedin Fig. 10.
index of the formulation is calculated by iteration.At the beginning no polymerization occurs. There-fore the refractive index of the formulation is equal tothat of the monomer at n � 1.48 �step 0�. As thetime increases, the received energy increases. Thereceived energy Er is calculated as the sum of smallamounts Era of energy. At each iteration over t, Erais added to Er
Er � �0
t
�W� x, z, t���2dt�,
which constitutes steps 3 and 4. Modifications �dn�of the refractive index of the medium appear gradu-ally, following the function F represented in Fig. 1�steps 5 and 6�. The analytical form of F that wasused is
F�Er� � dn�Er� � dnmax�1 � exp�Eth�Erp �� ,
Er � Eth, (8)
where dnmax � 0.04 and p represents the slope of thefunction at the beginning of polymerization. In thecalculations that we present here p � 1. This Ffunction has been taken to fit the empirical curve thatgives the degree of cross-linking as a function of thereceived energy.
The threshold energy Eth is a function of the dis-tance to the interface. This function represents theinfluence of oxygen diffusion on the polymerizationprocess. The closer we are to the interface, the moreenergy is needed for polymerization. The Eth func-tion, depicted in Fig. 12, was chosen as follows:
Eth� z� � 1 � 3 loge�1 �zd� , z � d. (9)
Here d is the height of the drop ��30 �m�. Thisfunction has been taken in accord with experimentalresults relating the polymerized thickness to the re-ceived energy of a polymer film.15 At each iterationthe BPM is performed on the modified medium.
Results of the calculations are presented in Fig. 13.The first column represents the refractive-index dis-tribution. The black areas are polymerized, andtheir refractive index is equal to 1.52; the white areascorrespond to unpolymerized formulation �refractiveindex of 1.48�. Note that the black area also corre-sponds to the part that remains after rinsing withmethanol. This part is directly related to the shapeof the polymer tip obtained in the real experiment.In the first column the black outlines represent theline K�W�x, z; t � 0��2t � Eth. In other words, theseblack outlines represent the theoretical shape of the
Fig. 13. Results of the calculation described in Fig. 10. Column1 represents the refractive-index distribution plotted as a functionof the exposure time t. The black outline represents the theoret-ical shape of the polymerized part if the modification of the refrac-tive index is neglected. Column 2 shows the received energy Er.The white outlines correspond to the isoenergetic lines Er � 1 andEr � 10. These outlines are also represented in gray in column 1.If the threshold energy Eth is not a function of the distance to theinterface, i.e., if oxygen diffusion is neglected, the shape of the
obtained tip must fit within the white outlines shown in column 2.Column 3 represents the instantaneous field intensity within thematerial, i.e., the result of the calculation of the beam propagationthrough the refractive-index distribution represented in column 1.
polymerized part if both the modification of the re-fractive index and the variation of the threshold en-ergy are neglected.
In the second column of Fig. 13 the received energyis presented. The white outlines correspond to theisoenergetic lines Er � 1 and Er � 10, as shown.These outlines are also represented in gray in thefirst column. If oxygen diffusion is not taken intoaccount the polymerization has to start as soon as thereceived energy passes over Er � 1. A comparison ofthe curve for Er � 1 and the shape of the tip �seecolumn 1� shows the influence of oxygen diffusion onthe shape of the tip extremity. Indeed, if the thresh-old energy is not a function of the distance to theinterface �i.e., if the oxygen diffusion is neglected�, theshape of the tip must fit the white outlines shown incolumn 2.
The third column of Fig. 13 represents the instan-taneous field intensity within the material, i.e., theresult of the calculation of the beam propagationthrough the refractive-index distribution representedin column 1. The images in column 3 illustrate theself-guiding character of the light. One can also seethe formation of nodes in the beam where the lightconcentrates. These nodes also exist on the tip ob-tained �column 1�.
This phenomenon of node formation has been ob-served on some fabricated tips, such as the one shownin Fig. 14. If the BPM is performed over longer dis-tances, one can see that the beam rapidly becomesself-guided and self-focused. The self-focusing effectrevealed by Figs. 13 and 14 confirms the observationmade by Kewitsch and Yariv16 five years ago. InRef. 16, the authors demonstrated, by using a UV-sensitive formulation, that optical beams can be self-focused and self-trapped on photopolymerization.This phenomenon is typical of nonlinear optical me-dia and reminds us of spatial soliton propagation in aphotorefractive medium.17 In Fig. 13 it should benoted that a rounded tip extremity is obtained onlywhen the threshold energy varies as a function of thez coordinate. Then, as can be seen from Fig. 13, theradius of curvature increases progressively as a func-tion of time, as was observed experimentally �Fig. 4�.
In summary, the above calculations, even if theyare rather qualitative, show that, as a result of the
modification of the refractive index of the mediumduring the polymerization process, no enlargement ofthe tip is expected. Moreover, the tip extremity issmoothed owing to oxygen diffusion from the inter-face that results in a z dependence of the thresholdenergy.
5. Perspectives and Potential Applications
In the near future, we plan to undertake a study ofthe optical properties of a polymer element that isintegrated on a single-mode fiber. The element ismade up of two parts: the tip body and the tip end.
A. Tip End
Considering the smooth hemispherical shape re-vealed, for example, in Fig. 3�b�, we expect that thetip extremity should act as a microlens. Accordingto the simple theoretical considerations proposed inRef. 6, the focal length and the minimum spot size ofthe microlens shown in Fig. 3�b� probably do not ex-ceed 3 �m and 1 �m, respectively. However, wehave to wonder what the term microlens means whenthe radius of curvature is not very large comparedwith the wavelength of the light, as is the case shownin Fig. 3�b�. In this case the tip end should actrather as a scattering Mie particle.18
B. Tip Body
As a preliminary study, we recently coupled laserlight �� � 542 nm� into a fiber through a cleaved endto observe, by optical microscopy, the output lightthat issues from the tip �Fig. 15�. Three facts werenoted:
�1� The full angle of the divergence of the lightemerging from the tip ��60°� is much larger thanthat at the output of the cleaved fiber ��14°�. Thisresult lets us hope that a high potential exists for thetip’s coupling with an incident diode-laser beam.1–3
Fig. 14. Electron micrograph of a polymer tip fabricated over thecore of a single-mode fiber. The tip body shows a node, as pre-dicted by the numerical results of Fig. 13.
Fig. 15. Schematic of the visual observation of the light emergingfrom the tipped fiber.
�2� The light emerges from only the tip extremity.This outcome means that, in the polymerized part,total-internal-reflection �polymer–air� conditions aremaintained from the fiber core to the tip end.
�3� The total intensity of the light emerging fromthe tip is equal to that which would issue from asimple cleaved optical fiber, demonstrating that thepolymer tip does not induce any appreciable opticalenergy loss.
Hence the tip would act as an optical waveguide withan approximate length of 30 �m, a core diameter of2at � 3.5 �m, a core refractive index of n1 � 1.52�very near to that of the initial fiber�, and a claddingrefractive index of n2 � 1 �air�. Thus the tip’s Vparameter is V � �2at����n1 � n2�1�2 � 25. At thishigh value of V, one expects that propagation insidethe polymer tip could be multimodal. However, be-cause the tip length is extremely small, there is ahigh probability that no mode coupling can occur and,consequently, that only the fundamental mode can beselectively excited inside the tip by the LP01 modethat propagates in the fiber core, provided that theaxis of the core fiber and that of the tip are perfectlyaligned �Fig. 3�a� indicates that this is exactly thecase . In any case far-field observations cannot com-prise a sure test for deciding if the tip is single modeor multimode. By scanning the near field of the tipend along its diameter by using scanning near-fieldoptical microscopy19,20 �SNOM�, we can ascertain themodal characteristics of the tip.
C. Possible Applications
The elements fabricated by the polymerization pro-cess are clearly attractive for optical applications.Two examples of original results that were obtainedwith polymer-tip fibers are presented below.
1. Optical Submicrometer-Resolution ProfilometryWhen regarding, for example, Fig. 4�a�, one expectsthat the single-mode tipped fiber can be used as alight collector for optical imaging with submicrome-ter resolution. This functionality was demonstratedby the experiment depicted in Fig. 16. A laser diodeworking in the spontaneous-emission mode was in-vestigated by use of a single-mode tipped fiber similarto that shown in Fig. 4�a�. The tipped fiber wasscanned above the laser diode during operation.The polymer tip was in contact with the crystal facetwhile the signal was collected at the other extremityof the fiber, where a photomultiplier was placed.
The laser diode studied is a commercial laser�SHARP, Model LT020MC� that has a V-grooved sub-strate internal stripe structure, a threshold current of41 mA, � � 780 nm, and a maximum optical power of3 mW. The laser is composed of a double heterojunc-tion �Ga1�xAlxAs�Ga1�yAlyAs�Ga1�xAlxAs, with y x 0.37�. The active layer is 200 nm wide.
For the experiment the pumping current was ad-justed to 30 mA. Figure 17 shows the 8 �m � 8 �moptical images that were obtained of the laser diode inoperation when scanning the tipped fiber at variousdistances from the diode. We note that the smallerthe distance, the more visible the emitting activelayer. During this experiment no lens effect �no fo-cal plane� was observed. This fact is not surprisingbecause the radius of curvature of the tip extremity isroughly equal to the wavelength of the light. In thiscase it seems difficult to use the concept of “lens.” Infact, the polymer tip seems to act as a probe for theSNOM18: The tip end acts like a Mie particle thatscatters the light in the vicinity of the sample. Thescattered light is guided first by the tip body and thenby the optical fiber and is detected at the other fiberextremity. The width at Imax�e2 of the light-emitting active layer revealed in Fig. 17�c� is 1.4 �m,which is similar to that recently reported by Lienau etal.21 who studied the same type of laser diode by
Fig. 16. Diagram of one use of the single-mode tipped fiber: im-aging a laser diode in operation. The image is formed by SNOMof the tip above the laser diode parallel to the crystal’s output facet.
Fig. 17. Results of the experiment depicted in Fig. 16. Shown are 8 �m � 8 �m optical images of the laser diode working inspontaneous-emission mode for various tip-to-diode distances: �a� �2.5 �m, �b� �1 �m, �c� �0.1 �m.
SNOM. For comparison, the experiment was alsoimplemented by use of a single-mode fiber without apolymer tip. In this case, even with the fiber in con-tact with the laser diode, the optical image wasalways circular in shape with a diameter of approx-imately 3 �m, corresponding to the fiber-core diam-eter. In conclusion, although the exact process oflight detection has to be studied, the polymer tipimproves the confinement of light that is permittedby the fiber and acts as a submicrometer-sized opticalprobe.
2. Selective-Mode Excitation of MultimodeOptical WaveguidesAs was pointed out in Subsection 3.C, the LP modesof the 4.5-�m core-radius fiber can be selected by useof mechanical strains. This selection mode is noteasy and is rather uncertain. However, after thepolymer mold �corresponding to the preselectedmode� is fabricated it can be used to couple the cor-
responding mode with a strong probability. Thisfact was demonstrated by the experiment depicted inFig. 18. A LP21-mode polymer tip �similar to thatshown in Fig. 7�d� attached to a 4.5-�m core-radiusfiber was illuminated by a slightly converging laserbeam that issued from an objective lens �NA � 0.12�.This illumination was carried out without particularprecautions. The distribution of light issued fromthe other extremity was characterized by a CCD cam-era. By adjusting the tilt ��x, �y� of the fiber, weobtained the results shown in Fig. 19, which presentsthe images provided by the camera for various fibertilts. We note that the LP21 mode was easily excitedin the fiber. The fiber length was approximately200 m. The maximum signal �Fig. 19�a� corre-sponds to a specific fiber tilt for which the optical pathis suitable for the four peaks of the polymer element.Indeed, if a required mode has to be obtained thecoupling efficiency � can be evaluated by the well-known overlap integral
� �
����
�
E1 E2dxdy�2
���
�
�E1�2dxdy ���
�
�E2�2dxdy
, (10)
where E1 and E2 are, respectively, the complex am-plitude of the mode to be coupled and the complexamplitude of the incident field. In Eq. �10�, x and yare the coordinates in a plane that is perpendicular tothe fiber axis. The value of � is high if E1 and E2have similar spatial distributions in both intensityand phase. When illuminating the end of the poly-mer tip, the light is guided by the four peaks thatconstitute the tip, and it emerges into the fiber corewith an intensity distribution similar to that of theLP21 mode �that is, similar to that of Fig. 7�c� . Nev-ertheless, the phase shift between two adjacentlobes of the LP21 mode is, a priori, not permitted bythe polymer tip. This specific phase shift can, how-ever, be obtained for a specific tilt ��ox, �oy� of the fiberwith respect to the incident beam, which can roughly
Fig. 18. Diagram of one use of the LP21-mode tipped fiber: se-lective mode coupling. The tip is illuminated by a slightly con-verging Gaussian beam. The intensity distribution at the fiberoutput is recorded by a CCD camera. The tilt ��x, �y� of the fibercan be adjusted.
Fig. 19. Results of the experiment depicted in Fig. 18. The images reveal that the LP21 mode was easily and selectively coupled into themultimode fiber. Each image corresponds to a fiber tilt ��x, �y� with respect to the incident beam. �a� Image obtained for optimal tilt:the coupling is strong. �b�, �c� Images obtained with nonoptimal tilts: the coupling efficiency is not strong, but the LP21 mode ismaintained.
be viewed as a plane wave at the focus of the 0.12-NAobjective lens. In the case of Fig. 19�a�, ��ox, �oy� wasachieved, and the LP21 mode was strongly coupledinto the fiber �the CCD camera has almost overload-ed�. In the case of Figs. 19�b� and 19�c�, the tilt wasmodified with respect to ��ox, �oy�, and the value of �decreased. However, the LP21 mode was main-tained. This method thus provides a good way toeasily make a multimode fiber work in a preselectedsingle-mode regime.
6. Conclusions
In conclusion, we have presented both a new appli-cation of free-radical photopolymerization and anew use for the formulation introduced in Refs. 8 and9 by reporting a simple method of producing amicrometer-sized polymer tip on the end of bothsingle-mode and multimode optical fibers. Themethod has been demonstrated to be very flexible andreproducible. Tip formation is based on the gradualgrowth of an optical waveguide in the liquid formu-lation just above the fiber core. This process hasbeen confirmed by a numerical calculation consistingof an iterative BPM in a medium whose the refractiveindex is time varying. This calculation has alsoshown that the influence of oxygen on the polymer-ization process can explain the shape of the tip ex-tremity.
Uses for the obtained components have been pro-posed and demonstrated by preliminary experimen-tal results. In the near future, we intend to extendour BPM to 3-D modeling and to study in detail theoptical properties of the fabricated components.Moreover, we plan to both propose new applicationsand put those applications fully into practice.
The authors thank G. Wurtz for his help in record-ing the images of Fig. 17, P. Chavel for providingpapers about the BPM, and D. Pagnoux and N. Fres-sengeas for fruitful discussions.
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