24 OPTICS LETTERS / Vol. 21, No. 1 / January 1, 1996

Self-focusing and self-trapping of optical beamsupon photopolymerization

Anthony S. Kewitsch* and Amnon Yariv

Department of Applied Physics, California Institute of Technology, 128-95, Pasadena, California 91125

Received October 2, 1995

We demonstrate theoretically and experimentally that optical beams are self-focused and self-trapped uponinitiating photopolymerization. This unique nonlinear optical phenomenon is dependent on the opticalexposure and produces permanent index-of-refraction changes larger than 0.04. The resulting nonlinear waveequation is shown to be nonlocal in time and displays self-trapped solutions only for suff iciently low averageoptical intensities. 1996 Optical Society of America

In this Letter we propose and demonstrate a newmechanism for self-focusing and self-trapping basedon photopolymerization. Photopolymerization pro-duces a permanent index change that is a function ofabsorbed optical energy rather than of intensity and islarger than the Kerr index change by orders of mag-nitude. Although Kerr solitons require an intensityof megawatts per square centimeter,1 we show thatlight beams can be self-trapped in photopolymers onlyfor sufficiently low average intensities. The responsetime of the Kerr nonlinearity is on the order of fem-toseconds, and the nonlinear optical response time ofphotopolymerization increases from milliseconds tominutes as the reaction proceeds. Like the Kerr effect,photopolymerization is a local phenomenon in space,for the index change at any location depends only onthe light intensity at that same location. However,the index change is nonlocal in time, for it depends onthe history of the optical electric field at all earliertimes. We show that this leads to a fundamentallynew form of the nonlinear wave equation that displaystransient, self-guided solutions.

The propagation of light in a photopolymer is de-scribed by the nonlinear wave equation for the opticalelectric field. The nonlinearity arises from the depen-dence of the index of refraction and absorption on opti-cal exposure. The index of refraction of the monomerduring cross-linking is measured on an Abbe refrac-tometer at 589.3 nm. For a diacrylate and triacry-late photopolymer, the maximum index changes are0.043 and 0.028, respectively. Figure 1 illustrates themeasured evolution of the index of refraction of the di-acrylate photopolymer. The index variation with ex-posure displays three distinct responses. During thefirst stage, for low exposures, the polymer chains aresmall (,10 monomer units). In addition, radicals arescavenged by the highly reactive oxygen molecules ab-sorbed by the photopolymer. The bonding of two rela-tively small molecules induces only a small density andindex-of-refraction change, corresponding to the ini-tial f lat induction period in Fig. 1. During the sec-ond stage, large polymer molecules are formed, andcross-linking between adjacent chains draws entirepolymer backbones together, resulting in a large in-dex change proportional to the density change. Unlikethe Kerr effect, the index response is delayed by 0.01–

0146-9592/96/010024-03$6.00/0

1 s relative to the illumination. The third stage ex-hibits saturation of the index change as the polymeri-zation reaction approaches completion. At this finalstage the diffusion constant of monomer radicals re-sponsible for chain building has decreased dramaticallyto a value as small as 10210 cm2 s21 (Ref. 2) so that sec-onds to minutes are required for the index change to de-velop. The largest index changes within the polymerare expected to be less than or equal to 0.15,3 well inexcess of traditional nonlinear optical phenomena suchas the Kerr and photorefractive effects, which displayindex changes of less than 1023.

Experimental results such as Fig. 1 for a wide rangeof acrylate photopolymers indicate that the indexresponse to an optical field amplitude E is of the form

Dn0sx, y, z, td ­ Dn0o

(1 2 exp

"2

1Uo

Z t2t

0jEst0dj2dt0

#),

(1)

where t is the monomer radical lifetime. The expres-sion for absorption photobleaching is approximately

Dn00sx, y, z, td ­ Dn00o exp

"2

1Uo

Z t

0jEst0dj2dt0

#. (2)

Uo is the critical exposure needed to induce polymeri-zation. Note that the real part of the index response

Fig. 1. Measured change in index of refraction (at589.3 nm) of the diacrylate photopolymer under UV expo-sure at 325 nm. The fitted curve is given by Eq. (2) withUo (in energy units) equal to 2.68 mJ cm22.

1996 Optical Society of America

January 1, 1996 / Vol. 21, No. 1 / OPTICS LETTERS 25

lags behind the application of the optical field by a timedelay t. This index change develops once the photo-generated radical diffuses to an appropriate monomerunit and induces cross-linking, rather than immedi-ately after the generation of radicals. However, theabsorption change associated with photoinitiator pho-tocleavage occurs within several femtoseconds. Onecan reduce or even eliminate the induction period ap-parent for small exposures in Fig. 1 by purging themonomer of oxygen radicals. The experimental datathen display good agreement with Eq. (1).

Several qualitative predictions regarding the evo-lution of the light-induced waveguide can be deducedfrom Fig. 1. At early times, the induction periodand the delay in the material response impede self-trapping. The induction period causes exposurethresholding. Polymerization is initiated first in re-gions of intense illumination, so the lateral dimensionof the waveguide is smaller than that of the opticalbeam. The lenslike index profile focuses primarilythe central part of the beam. At later times the indexincreases linearly with exposure and the index dis-tribution becomes a faithful replica of the transverseintensity profile. Self-focusing is strong, and thewaveguide extends through the medium. For largeexposure the index change saturates at the centerof the waveguide and self-focusing is diminished.At this stage the waveguide becomes essentially astep-index fiber. Of course, for very large exposuresthe entire liquid volume will polymerize because thetails of the optical field extend beyond the core and intothe uncured photopolymer. Although not treated byEqs. (1) and (2), the scatter of light and the diffusionof free radicals beyond the illuminated region furthercontribute to the increase in waveguide diameter andthe termination of guiding.

The guidance condition for an optical waveguide de-pends on the index change between the solid polymercore and the liquid polymer cladding. For a typicalindex change of 0.04, the smallest stable diameter ofthe self-trapped beam is expected to be 0.6 mm. Anadditional and unique condition for self-trapping inphotopolymers is that the average optical intensitymust be below an approximate threshold value givenby Iave , Uoyt, where Uo is the critical exposure neededto cure the photopolymer and t is the monomer radicallifetime. For a typical photopolymer composition (pho-toinitiator concentration 0.005 wt. %, Uo ø 1 J cm22,t ø 100 ms), the average intensity must be less than10 W cm22. The optical nonlinearity is unique in thatit occurs only for low average optical intensity. Intu-itively, if intense illumination were used, a large num-ber of radicals would be produced in a time that isshort compared with the radical lifetime t. The pho-topolymer would then cure completely in the illumi-nated region, before the optical beam experienced anindex change.

Analytical solutions to the nonlinear wave equa-tion with an index-of-refraction response given byEqs. (1) and (2) are obtained only for overly simplis-tic approximations. We therefore analyze the propa-gation of the optical wave through the photopolymernumerically by solving the complete nonlinear waveequation. The intensity profile is computed by the

beam-propagation method,4,5 incorporating the trans-parent boundary condition6 and the measured indexevolution. Typical photopolymer compositions used inthese experiments have low optical absorption asso-ciated with the photoinitiator, so photobleaching de-scribed by Eq. (2) can be ignored.

Two-dimensional numerical simulations exhibit self-trapped solutions to the nonlinear wave equation forlow average optical intensities. Figure 2 illustratesthe electric field amplitude of the self-trapped Gauss-

Fig. 2. Numerical simulations of beam propagation inphotopolymer in time steps equal to 5tc. Left-most simu-lation, t ­ 0; right-most simulation, t ­ 15tc. The ver-tical scale is 103 wavelengths; the horizontal scale is10 wavelengths.

Fig. 3. Experimental setup for self-focusing and self-trapping of optical beams. The input power is approxi-mately 5 mW at 325 nm.

Fig. 4. Experimental demonstration of self-focusing uponphotopolymerization. Left-most simulation, photopolymerat t ­ 0; right-most simulation, photopolymer at t ­30 s. The horizontal scale is 300 mm; the vertical scaleis 1 mm.

26 OPTICS LETTERS / Vol. 21, No. 1 / January 1, 1996

Fig. 5. (a) Series of fibers grown by self-trapping of theUV beam; the nominal diameter is 10 mm. (b) Three-dimensional fiber lattice with 50-mm-diameter filamentsof 5-mm length, fabricated within a photopolymer-f illedcuvette. (c) Array of fibers of 200-mm diameter and10-mm length.

ian beam in a photopolymer liquid. The waist of thebeam is located at the input surface of the photopoly-mer. The beam width at the input is equal to twowavelengths, and the propagation distance is 500 wave-lengths. Note that the horizontal scale is magnifiedby a factor of 10 compared with the vertical scale, to im-prove the transverse resolution. The left-most simula-tion illustrates the evolution of the beam through thehomogeneous photopolymer before any index changesare induced. The time steps from the left to the rightare in units of 5tc. tc is a parameter chosen to equalthe time needed to attain the critical exposure Uoat the location of maximum intensity of the opticalbeam. Self-trapping of the optical beam is apparentafter an exposure of 15tc. The weak oscillations inthe beam diameter evident at 15tc are typical for self-trapped beams.7 The background absorption lengthis 103 wavelengths, so self-trapping terminates in ap-proximately this distance.

We have experimentally observed self-focusing ina liquid diacrylate monomer using the experimen-tal setup of depicted in Fig. 3. The input beam is a1-mW, TEM00 mode at 325 nm with a 100-mm beamdiameter at the entrance face. The left-most simu-lation in Fig. 4 illustrates the beam profile at t ­ 0inside the liquid photopolymer. Time increases fromleft to right. The right-most simulation in Fig. 4 istaken after approximately 30 s of continuous illumi-nation. The position of the beam waist moves closerto the input plane as photopolymerization proceeds, amanifestation of self-focusing.

We observe dramatic self-trapping of optical beamsover distances in excess of 3 cm and diameters of10–50 mm by exposing the polymer to a sequence ofshort exposures separated by a diffusion time. Fig-ure 5(a) shows a series of solid polymer fibers 10 mmin diameter and 3 mm in length inside a diacry-late photopolymer. Since the solid is denser than theliquid surroundings, the structures bend downwardfrom the entrance face of the cuvette after severalhours. The waveguides can be steered during for-mation by introduction of an asymmetry across thetransverse intensity profile of the beam. Each fiber isexposed 30 times for 1y16 s, with a delay of 1–10 s be-

tween successive exposures to ensure that the averageintensity is below the threshold value given above. Anarray of fiberlike structures can be fabricated to pro-duce a three-dimensional microlattice [Fig. 5(b)]. Thediameter of each solid filament is 50 mm, spaced byapproximately 500 mm. The beam is guided over adistance greater than 10 mm. Regular arrays of postsof 200-mm diameter and 10-mm length have also beenfabricated in an epoxy–triacrylate photopolymer, as il-lustrated in Fig. 5(c). Each post is formed by a single1y8-s exposure, so the average intensity is too large toexhibit trapping. Hence, the diameters of the wave-guides are significantly larger.

The diameters of the fibers in Figs. 5(a) and 5(b)remain constant well beyond the confocal parameterof 122 mm for a Gaussian input beam with a 3-mmwaist. This is a signature of self-trapping. The di-ameter of the guided beam is highly dependent on thecomposition of the photopolymer, its oxygen content,and the optical intensity. The self-trapped beam isfound to survive over a distance approximately equalto the absorption length. These findings show goodqualitative agreement with the numerical simulations.Note that the oscillations in waveguide diameter pre-dicted numerically are of sufficiently small amplitude(,1 wavelength) that we do not expect to resolve themexperimentally.

In summary, we have demonstrated theoreticallyand experimentally that optical beams are self-focusedand self-trapped upon photopolymerization. The re-sulting nonlinear wave equation is unique in that it isnonlocal in time and displays transient yet pronouncedself-trapped solutions. Another unique feature of thisnonlinear optical effect is its slow time response. Thisphenomenon belongs to an entirely new class of opticalnonlinearity observed only at sufficiently low averageoptical intensity. We believe that this nonlinear opti-cal effect may have fundamental technological applica-tions to microfabrication, in which photopolymers areubiquitous as photoresists for photolithography andmicromachining.

This work was supported by the Advanced ResearchProject Agency’s nonlinear optics (DSO) and lithogra-phy (MTO) programs and U.S. Army Research Officecontract DAAH04-95-C-0063.

*Permanent address, Arroyo Optics, Inc., 1646 17thStreet, Santa Monica, California 90404.

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