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Optical tweezing by photomigration ZOUHEIR SEKKAT1,2,3 1

Department of Chemistry, Faculty of Sciences, University Mohammed V, Rabat, Morocco Optics and Photonics Center, Moroccan Foundation for Advanced Science and Innovation and Research, Rabat, Morocco 3 Department of Applied Physics, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan ([email protected]) 2

Received 8 July 2015; revised 18 November 2015; accepted 25 November 2015; posted 25 November 2015 (Doc. ID 245337); published 6 January 2016

Photomigration in azo polymers is an area of research that witnessed intensive studies owing to its potential in optical manipulation, e.g., optical tweezing, the physical mechanism of which remains unsolved since its discovery about two decades ago. In this paper, a detailed theoretical study that reproduces the phenomena associated with photomigration is presented, including the physical models and the associated master equations. Polarization effects are discussed and analytical solutions are given to describe the steady-state and the dynamics of photomigration. Such a theory leads to new theoretical experiments relating material properties to light action. A photoisomerization force which is described by a spring-type model is introduced. This force is derived from a harmonic light potential that moves the azo polymer. This force is parenting to optical tweezers, but it is quite different in the sense that it requires photoisomerization to occur. The azo polymer’s motion is governed by four competing forces: the photoisomerization force, and the restoring optical gradient and elastic forces, as well as the random forces due to spontaneous diffusion. © 2016 Optical Society of America OCIS codes: (160.5335) Photosensitive materials; (160.5470) Polymers; (310.6870) Thin films, other properties; (140.3390) Laser materials processing; (140.7010) Laser trapping. http://dx.doi.org/10.1364/AO.55.000259

1. INTRODUCTION Light action on matter dates back to the end of the 19th century and the beginning of the past century where the role of momentum exchange and radiation pressure in moving matter were pointed out [1–3]. The minuteness of such forces put them aside of practical applications until 1970, with the advent of the lasers, when Ashkin showed that optical forces can alter the motion of small particles and atoms [4,5], and paved the way for laser cooling of atoms [6,7]. The seminal work of Ashkin eventually led to the development of a single-beam optical trap by the optical gradient force known as optical tweezers [8,9], and today they find applications in physics and biology owing to their ability to apply piconewton (pN) forces to micrometer-sized particles and simultaneously measure displacement in the nanometer scale [10,11]. Optical tweezers have been used for the study of molecular motors at the single molecule level [12,13], and colloids and mesoscopic systems [14,15], as well as polymers and biopolymers [16–20]. For recent reviews on optical trapping and manipulation of micro/ nanostructures see Refs. [10,11]. This paper is about optical manipulation in films of photosensitive polymers, namely, azo polymers, and we will show that in such materials, optical tweezing occurs by a new type of light forces, e.g., the photoisomerization force. Such a force requires the existence of a 1559-128X/16/020259-10$15/0$15.00 © 2016 Optical Society of America

photochemical process, e.g., photoisomerization, to proceed in contrast to the gradient and elastic forces (Fig. 1). This finding is due to an effort of modeling and explaining the phenomena associated with photoinduced molecular movement, e.g., photomigration, in azo polymers. Photoisomerizable polymers, namely, azo polymers, have attracted much attention because of their tremendous importance in a large spectrum of adjacent research fields, including photochemistry, polymer science and engineering, chemical engineering, optics and nonlinear optics, and they are entering the field of optical tweezers as nanomanipulating materials. The basic phenomena discovered in the field of azo polymers about two decades ago, including photoinduced mass movement of polymers, are summarized in a book published in 2002 [21]. Recent reviews targeted applications of light-triggered azopolymer movement [22,23]. Photoinduced patterns of surface deformations in azobenzene containing polymer films have attracted much attention because of possible applications in optical data storage and in nanofabrication, and it is well known that such patterns reflect the state of the incident light polarization and the light intensity distribution [24–45]. The photoinduced patterns are due to light-induced mass movement of the polymer chains which in turn is triggered by the photoisomerization of the azo chromophores.

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Fig. 1. Schematic of the forces (photoisomerization, optical gradient, and elastic) that may create motion of an azo-polymer particle which is subject to a focused light beam. The photoisomerization force occurs only when the photon energy of the beam hits a photoisomerization transition of the azo molecules in the particle. The latter is a part of a polymer film. The laboratory coordinates as well as the directions of light polarization and the forces are indicated.

The molecular machine, e.g., the azobenzene derivative that fuels the polymer motion has two geometric isomers, the trans and the cis isomers, and the isomerization reaction is a light- or heat-induced interconversion of the two isomers. The trans isomer is thermodynamically more stable than the cis isomer—the energy barrier at room temperature is about 50 kJ∕mol for the azobenzene—and generally, the thermal isomerization is in the cis→trans direction. Light induces isomerization in both directions. Photoisomerization begins by elevating the isomers to electronically exited states, after which nonradiative decay brings them to the ground state either in the “cis” form or in the “trans” form, the ratio depending upon the quantum yields (QYs) of the isomerization reaction. From the cis form, the chromophore comes back to the trans form by two mechanisms: the spontaneous thermal reaction and the reverse cis→trans photoisomerization [46,47]. Azo dyes change shape in the excited states, and photoisomerization to the cis form takes place either via rotation or inversion mechanisms [46]. It has been shown that azobenzene derivatives shake a thousand times between the ground state and the electronically excited states before isomerizing [48]. The trans-cis photoreaction lasts a few picoseconds [46], and the QYs of azo dye derivatives in polymers are, for example, for Disperse Red One (DR1) in films polymethyl methacrylate (PMMA), 0.11 for trans-cis and 0.7 for the cis-trans photoisomerizations [49]. DR1 chemically attached to PMMA is the most studied azo-polymer systems in photoinduced mass movement, and the QY values show that DR1-trans isomerizes to DR1-cis for about 10 absorbed photons and DR1-cis isomerizes back to DR1-trans for about one absorbed photon. Other azo dye chromophores present QY values in the range of 0.001 and 0.2 for the trans-cis and cis-trans photoisomerizations, respectively, showing a perpetual motion, e.g., springtype motion, of the azo molecule between the two-isomers, e.g., an elongated trans and a contracted, e.g., a globular, cis

Research Article isomer [48]. In photoinduced mass movement experiments in azo polymers, the chromophores density of which is typically of the order of 1021 per cubic centimeter, irradiation intensities, which are in the range of a few tens of milliwatts per centimeter square, impose 10 incident photons per azo chromophore per second. The azo chromophore is in permanent contraction and elongation motion, e.g., a permanent spring-type motion. When the irradiation light is polarized, the azo molecules are redistributed perpendicular to the irradiation light polarization after light absorption [47–52]. Photo-orientation results from photoselective isomerization. For one photon isomerization, photoselection imposes a reaction proportional to cos2 θ where θ is the angle between the direction of light polarization and the transition dipole moment (TDM) of the chromophore. The term cos2 θ defines the symmetry of the light polarization with respect to the absorption of the chromophores, e.g., molecules that have their TDM oriented along the irradiation light polarization have the highest probability of isomerization and those that are oriented perpendicular to the excitation light polarization have the lowest probability of isomerization. Photoselection burns a hole into the molecular orientational distribution, and the long molecular axis of the azo chromophore fades from the exciting light polarization due to rotation during photoisomerization, thereby inducing molecular photo-orientation. A surface relief, which is due to a photoinduced mass movement of the azo polymer, is fabricated by an interference pattern of light. The polymer moves from high- to low-intensity regions in the direction of light polarization, and the trans-cis photoselective isomerization plays an important role in the deformation process. Several authors proposed theoretical models to describe photoisomerization-induced mass movement in azo polymers [53–68]. For example, Kumar et al. attributed photomigration to dipoles interacting with the gradient of an optical electric field [53,54], and Barett et al. introduced a model based on light-induced pressure [55]. Nunzi et al. introduced a diffusion model based on a random walk motion [56]. This type of model was also proposed by Bellini et al. [57]. Pedersen et al. proposed a model based on a mean-field theory [58], and Sumaru et al. proposed a fluid mechanics model [59,60], and Juan et al. proposed a stochastic model [61,62]. Galinski et al. proposed a model based on phase separation [63], Toshchevikov et al. proposed a microscopic model that accounts for the internal structure of polymer chains [64], Ilnytskyi et al. performed a molecular dynamics study based on molecular orientation and orientational hole burning [65], and Voit et al. proposed a model based on thermo-diffusion [66,67]. Yet, a theoretical model that comprehensively and clearly describes the phenomenon of photomigration, with an appropriate formalism, taking into account the combined aspects of photochemistry and photoselection and polymer science and molecular mobility, is needed. This is what is investigated in this paper. I will show that the mechanism of photomigration is based on a photoisomerization force which is quite different from the optical gradient force. In fact, the latter is competing with the photoisomerization force to restore the polymer to its initial position. In other words, in photomigration, the optical gradient force is a restoring force together with the force due to the polymer’s elasticity (vide infra).

Research Article From an experimental point of view, photoisomerization was shown to enhance molecular mobility far below the glass transition temperature (Tg) of azo polymers in the 1990s [51,52,69–72], and considerable exploration of sub-Tg photoinduced molecular movement was performed especially targeting polymer structural effects, including Tg, the free volume and free volume distribution, the mode of the attachment of the chromophore to a rigid or flexible chain, the molecular weight, and so on [73,74]. Light-induced mass movement, e.g., work generation by light, in azo polymers, i.e., surface relief gratings (SRGs) [33,34], triggered many studies to understand the mechanism of polymer migration, and most of the studies have focused on SRGs which are fabricated by the interference pattern of two coherent laser beams [33–39]. Recently, single-beam-induced deformations of the azo polymers shade light onto the mechanism of photoinduced molecular migration [30,40–45]. Besides light-induced mass movement, photoinduced contraction and bending can also generate work from azo polymers. Such light-sensitive systems may be referred to as light-energy transducers, and a recent review perspective about light-to-work transduction in photosensitive systems was recently published [75]. In the contraction and bending experiments, free-standing films of azo polymers bend reversibly upon light action because only the top fraction of a few micrometers, and up to millimeters, thick polymer is softened by light that penetrates a few hundreds of nanometers in the film, thereby leaving the rest of the film unaffected by light—a feature which creates a gradient of mechanical properties in the film in the direction perpendicular to the film thickness. Such a chromophore for reversible shape change is the rhodopsinretinal protein photoswitch system that enables vision. It can be found in nature. Perhaps the best artificial mimic of this strong photoswitching effect is that of azobenzene derivatives. An interesting phenomenon of light-driven motion of matter by photoisomerization was reported by Zyss et al. for azobenzenecoated nanoparticles in an aqueous medium [76]. The nanoparticles were propelled toward the light intensity minima by the mechanical work supplied by photoisomerization. In this paper we discuss light-triggered molecular migration, referred hereto as photomigration, in azo polymers and we will introduce the photoisomerization force as parenting to optical tweezers to describe photoinduced molecular migration.

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azobenzene derivative (vide infra). Such a photoinduced cyclic movement of contraction and relaxation, which can be modeled by a spring-type movement, leads to a translational motion in any direction of the 3-dimensional space. When a contracted spring is released, it changes position if there are no constraints at either one or both ends of the spring—a feature which I believe is at the origin of the photoinduced azo-polymer movement (Fig. 2). Such a motion is parenting to the wormlike motion proposed by Nunzi et al. [56], and the latter motion needs support points to proceed. The translation is peaked in the direction of light polarization because of anisotropic optical tweezing, and it is due to the driving potential of the migration of the particles. In the case of linearly polarized light, the photoisomerization potential is U photo
x  N δf
1 − ασ T ϕT C  ασ C ϕCT gII
x:

(1)

U photo
x ≡ U is the potential acting on the particle (expressed in joules, J), x (expressed in cm), σ T and σ C (expressed in cm2 ∕molecule) are the isotropic absorption cross sections, at the irradiation wavelength, of the trans and cis azo molecules of the particles, respectively. The cross sections are proportional to the isomers molar absorptivities. ϕT C and ϕCT are the quantum yields of photoisomerization. They represent the efficiency of the trans→cis and cis→trans photochemical conversion per absorbed photon, respectively. I
x ≡ I is the light intensity in the direction of the light polarization. I is normalized by the incident light intensity I 0 which is the actinic flux density, e.g., the energy crossing a unit area per unit time (expressed in J cm−2 s−1 ). I  I 0 t is the energy dose per unit area (expressed in J cm−2 ). N is the density of the absorbing particles (expressed in molecules ⋅ cm−3 ), α is the extent of the cis isomer, and δ the absorption path length (expressed in cm). It is worth noting

2. PHENOMENOLOGY OF PHOTOMIGRATION A. Photoisomerization Force

Consider that a collection of spherical Brownian particles that are weakly interacting with each other, all having equal size, are suspended in a soft material (in general, a viscous fluid or a polymer that behaves as a viscous liquid such as liquid crystalline and low molecular weight flexible polymers). Experimental work suggests that azo polymers exhibit visco-fluidic motion when the azo molecules undergo photoisomerization [27–29]. The particle, which we assume to be spherical for simplicity and which can be viewed as the azobenzene derivative molecule surrounded by the nanocogent environment of the polymer segments, contracts and relaxes to its original shape under the perpetual motion of cyclic photoisomerization of the

Fig. 2. (Top) Schematic of photomigration in a cross-sectional view of an azo-polymer film. (Bottom) Trans-cis isomerization of azobenzenes. The elongated and contracted springs refer to the trans- and cisazobenzene, respectively. The azo-polymer film is indicated by the red color, and the straight arrow refers to the trans to cis isomerization, while the curved arrows refer to the cis to trans isomerization. The trans isomer is removed from the center, e.g., the location of the irradiation light, and moved by the photoisomerization force to the sides where no light exists.

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that molecular reorientation effects are not to be taken into account in the expression of U because molecules are removed from their initial position by migration when photoisomerized. However, the orientation of the molecules may change when they migrate from bright to dark areas by photoisomerization. The light potential exerts a photoselective gradient force, referred hereto as the photoisomerization force ∂U : (2) F photo  − ∂x The photoisomerization force described by Eq. (2) opposes the optical gradient force (vide infra). Equation (2) shows that F photo (expressed in J cm−1 ) is proportional to the gradient of light intensity ∂I (3) F photo ∝ ; ∂x

where kphoto

∂I F photo  −k photo ; ∂x (expressed in J cm−2 ) is given by

kphoto  N δf
1 − ασ T ϕT C  ασ C ϕCT gI:

(4) (5)

When the irradiation light intensity is weak, α ∼ 0, and kphoto reduces to kphoto  N δσ T ϕT C I:

(6)

A power of the gradient of the light intensity may be introduced to account for deviation from linearity, e.g., for the regime of strong photomigration, so that the general expression of the photoisomerization force will be
β ∂I ; (7) F∝ ∂x where β is a constant. Such an expression has some experimental foundation [30], but for simplicity, I consider β  1. It can be seen from Eq. (1) that photomigration proceeds with light intensity and with both, e.g., the forth σ T ϕT C and back σ C ϕCT photoisomerizations, or only the forth photoisomerization taking place. The material, e.g., azo polymer, is polarization sensitive, and photoselection imposes the direction of the polymer motion. Molecules which are oriented along the polarization of the actinic light present the highest probability of being photoisomerized, and the polymer is softened in the direction of light polarization and molecular movement proceeds in this direction. Such a motion can be along a line if the actinic light is linearly polarized, or a curved line, e.g., a circle or an ellipse, if the actinic light is circularly or elliptically polarized, respectively, or a helix for vortex-type irradiation (Fig. 3) [25,26,42]. B. Polarization Effects in Photomigration

Polarization effects in photomigration can be accounted for in the expression of the distribution of the light intensity in Eq. (1). Indeed, if the light intensity is linearly polarized, say in the x-direction, the electric field components of a plane wave at the film surface propagating perpendicular to the azopolymer layer are E x  E 0 , E y  0, and E z  0, where E 0 is the amplitude of the electric field of the incident light, with z as the direction of propagation (perpendicular to x), and y

Fig. 3. AFM images of the deformation induced by a tightly focused laser beam polarized (a) horizontally, (b) vertically, and (c) circularly, respectively, and (d) by vortex-type beam. Figures (a)–(c) and (d) are reproduced respectively with permission from [42] © 2007 Optical Society of America, and Macmillan Publishers Ltd: Ambrosio, Marrucci, Borbone, Roviello, Maddalena, Nature Comm., Vol. 3, 989 (2012) [25]. © 2012.

perpendicular to both x and z. In photomigration experiments, the azo-polymer layer is limited on one side by a semi-infinite glass substrate and on the other side by a semi-infinite air medium. So the light intensity at the polymer surface is exclusively x-polarized, e.g., I x . I x  jE x j2  jE 0 j2  I 0 , and the photoisomerization force is originating exclusively from the gradient of that component of the light intensity—a feature which importantly shows that the azo polymer proceeds in the direction of light polarization during photomigration due to the gradient of the light intensity in this direction F photo ∝ ∂I x ∕∂x. No gradients of light intensity exist in the y- and z-directions, thereby no photoisomerization force acts on these directions. That is why in surface relief experiments no photomigration motion is observed when the actinic light polarization is parallel to the interference pattern; F photo ∝
∂I y ∕∂y  0. Now, if the actinic x-linearly polarized light beam is focused, e.g., a Gaussian beam, some components of the electric field occur in the y- and z-directions as well, but those remain small compared to E x even if the beam is tightly focused by a high numerical aperture (N.A.) objective lens. Calculated distribution of squared electric field components created by a tightly focused linearly polarized (in the x-direction) laser beam shows that the maximum intensity corresponding to E z and E y are 7 and 200 times smaller, respectively, than E x for N:A:  1.4 and actinic light wavelength  460 nm with a refractive index of the medium surrounding the azo-polymer layer equal to 1.5 (Fig. 4, Top) [42]. In such experimental conditions only E x contributes appreciably to the photomigration motion because it corresponds to the dominant intensity gradient of the light. Photomigration created by a light which is linearly polarized in the direction z perpendicular to the azo-polymer layer, e.g., by the longitudinal component of the electric field E z , should proceed in the z-direction (F photo ∝ ∂I z ∕∂z). Such a phenomenon has been recently observed whereby the longitudinal field E z that creates longitudinal photomigration, e.g., the azo polymer grows perpendicular to the layer surface under E z (Fig. 5), was created by radial polarization focused by a 1.4 N.A. objective lens through an annular mask; the annular size of which corresponds to N.A. ∼1.0 to 1.2. Such a configuration leads to an E z inside the focused spot five times E x;y and it creates a Bessel beam which has a long focal length, thereby the light distribution does not spread inside the film, and the deformation pattern does not depend on the focus position. Bessel

Research Article

Fig. 4. (Top) Calculated distributions of squared electric field components created by a tightly focused linearly polarized laser beam. The components of the electric field of (a) E x , (b) E y , and (c) E z are shown. The polarization direction is X, Z is perpendicular to the film, and Y is perpendicular to both X and Z. The distribution was calculated assuming a refractive index of the surrounding medium equal to 1.5. (Bottom) A typical deformation pattern produced by E x (left) and the corresponding line plot (right). The direction of light polarization is indicated in the left figure; the position of the maximum population, e.g., the height, as well as the FWHM are indicated in the right figure by a vertical and a horizontal arrow, respectively. Reprinted with permission from Ref. [42]. © 2007 Optical Society of America.

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the polarization, the intensity of Eq. (1) is written as I  jE T j2  jE L j2 , and the photoisomerization force is due to the components of the intensity gradient from both the transverse and the longitudinal directions F photo ∝
ˆz ∂I z ∕∂z  ρˆ ∂I ρ ∕∂ρ. ρ is the radial transverse spatial position (parallel to the film plane) in polar coordinates, with ρˆ its unit vector, and zˆ is the unit vector of the z-axis. Vortex-beam irradiation leads to such a force [25,26]. For plane wave irradiation, I  I ρ  jE T j2 and F photo ∝ ∂I ρ ∕∂ρ. For tightly focused irradiation, Bessel beams generate both transverse and longitudinal components of the electric field, the squared modulus of which can be described by simple analytical expressions inside the polymer layer, e.g., by the zeroth- and first-order Bessel functions of the first kind [78]. Bessel beams can be radially or azimuthally or circularly polarized. jE L j2  0 for azimuthal polarization. From the above analysis, it occurs that the photoisomerization force is a photoselective gradient force. Such a force represents a new type of optical forces, and it is parenting to optical tweezers, and it requires the existence of a polarization-dependent gradient of light intensity and photochemical phenomenon, e.g., photoisomerization, to trigger molecular motion. The photoisomerization force is a repelling force and its direction corresponds to the direction of light polarization. I will go on to introduce the master equations of photomigration. C. Optical Gradient and Elastic Forces

beams are the nondiffracting solutions of the Maxwell equations [77]. It is instructive to study photomigration by focused light irradiation with a high aperture angle Bessel beam because it is possible to separate the effects of transverse and longitudinal polarizations on photomigration. When the actinic light has both transverse, E T parallel to the film surface, and longitudinal, E L  E z , components of

Fig. 5. (Top) AFM images of the deformation induced by a tightly focused laser beam polarized (left) longitudinally, e.g., by E z, and (right) laterally, e.g., E y , on a 60 nm thick azo polymer. The right figure is shown for comparison, and the direction of the light polarization is indicated. Note that photomigration occurs in the longitudinal and lateral directions for the left and right figures, respectively. (Bottom) Corresponding line scans of the deformation patterns. Reprinted with permission from H. Ishitobi, I. Nakamura, T. Kobayashi, N. Hayazawa, Z. Sekkat, S. Kawata, and Y. Inouye, ACS Photon. 1, 190-197 (2014) [30]. © 2014 American Chemical Society.

The light potential may also exert an optical gradient force, F gradient , that attracts the particle to or repels it from the bright area, depending on the refractive indices of the material np and the surrounding medium ns [8,11]. ∂I (8) F gradient  −kgradient ; ∂x where
 n3s n2p ∕n2s − 1 I : kgradient ∝ (9) 2 n2p ∕n2s − 2 0 In general, F gradient opposes F photo . It has been shown that, in an experimental configuration where ns was changed from 1 (air) to 1.47 (glycerin), with np  1.67, F gradient decreased by a factor of ∼2 and its direction was changed from attractive to repulsive—a feature which is consistent with Eq. (9) which glycerin yields F gradient ∼ −0.57F air gradient with the above values of ns and np . In this experiment, the surrounding medium is the medium which is on top of an azo-polymer film, which in turn is deposited on a glass substrate [30]. The optical gradient force has been observed in photosoftened azo polymers only in some extreme conditions, e.g., a strongly focused beam with the maximum intensity located in air not on the film surface [42] or with longitudinal polarization where both the optical gradient force and photoisomerization force have the same direction and they can interfere with each other in a constructive or destructive way [30]. Another restoring force due to the polymer’s elasticity can be taken into account in light-induced molecular migration, and under the weak force condition, F elastic is linear in displacement x

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F elastic  −k elastic x;

(10)

where kelastic represents the elastic properties of the polymer and it depends on the polymer molecular architecture, the mode of connection of the azo chromophore to the polymer chain, the molecular weight, the free volume distribution, and Tg. kelastic may be measured from photomigration experiments (vide infra). F elastic opposes F photo (Fig. 1). The elastic force is due to an elastic harmonic potential, as given by Hook’s law 1 (11) U elastic  k elastic x 2 : 2 The total force and potential acting on the particles read, respectively, as F⃗  F⃗ photo  F⃗ gradient  F⃗ elastic ;

(12)

U  U photo  U gradient  U elastic :

(13)

3. EQUILIBRIUM AND DYNAMICS OF PHOTOMIGRATION I approach photomigration of photoactive particles by the Brownian motion, due to random forces, of weakly interacting particles subjected to a light potential described by Eq. (13). Such a physical phenomenon can be described by the Smoluchowski equation for the dynamics of polymer solutions and suspensions, which is a phenomenological tool for describing the fluctuations of physical quantities. For example it can be used to describe the fluctuations of thermodynamic variables such as concentration. If Ψ
x; t is the probability distribution function, for non or weakly interacting particles, that a particular particle is found at point x at time t, the evolution equation for the normalized probability Ψ
x; t is written as reported by Doi and Edwards [79] as
 ∂Ψ X ∂ ∂Ψ ∂U k T  L  Ψ : (14) ∂t ∂x n nm B ∂x m ∂x m n;m For a system that has many degrees of freedom, x 1 ; x 2 ; …; x n ≡ fxg is the set of dynamical variables describing the state of Brownian particles. The coefficients Lmn are called the mobility matrix and may be obtained using hydrodynamics. U represents an external potential, e.g., given by Eq. (13) for example, which exerts a force F on the particle F n  −∂U ∕∂x n :

(15)

Since the movement proceeds in the direction of light polarization, Eq. (13) reduces to Eq. (16) with x being the direction of light polarization and molecular migration
 ∂Ψ ∂1 ∂Ψ ∂U  k T  Ψ ; (16) ∂t ∂x ξ B ∂x ∂x where kB is the Boltzmann constant and T is the absolute temperature. ξ specifies the response to the external force, and it is related to the quantity D which characterizes the thermal, e.g., random, motion of the particles by the Einstein relation [80] k T (17) D B : ξ The coefficient ξ is called the friction constant and its reciprocal 1∕ξ is called the mobility. ξ can be obtained from

hydrodynamics if the particle is large enough. For example, if the particle is spherical with a radius r, and the viscosity of the solvent is ηs , then the hydrodynamic calculations indicate [81] ξ  6πηs r:

(18)

The relationship of the force and velocity holds independently of the shape of the particle provided that the force is weak. Strictly speaking, ξ depends on light intensity since the particle changes shape upon cyclic photoisomerization of the azo molecules, but for simplicity, it is assumed to be constant for a given intensity. In other words, ξ
I, e.g., with light on, is smaller than ξ, e.g., with light off, e.g., the mobility of the particles is larger when the molecules are isomerized from an elongated trans isomer to a more globular cis isomer (r of the cis isomer is smaller than that of the trans isomer). Such a model has proven to be successful for the photoinduced mobility of azobenzene derivatives in polymers [71,72], and it gives the foundation for the concept of softening azopolymers by photoisomerization. Recently, manipulating the mechanical properties of azo polymers is researched intensively owing to the possibility of creating mechanical work by light [75,82,83]. The solution of Eq. (16) for free particles, e.g., U  0 is [79,84] 

x − x 0 2 −1∕2 ; (19) exp − Ψ
x; t 
4πDt 4Dt where x 0 is the position of the particle at time t  0. This is a natural solution for a Brownian motion due to random forces due to the incessant collision of the fluid molecules with the Brownian particles. The sum of the random forces is a stochastic variable, and if a Gaussian distribution is assumed for it to satisfy Eq. (16), then the distribution of x
t must be Gaussian. A. Equilibrium of Photomigration

For the situation where the Brownian particles are subjected to a light potential, an exact solution of Eq. (16) can be found in the case of weak forces where the gradient of light intensity is linear in x. In this case all of the potentials, e.g., photoisomerization, optical gradient, and elastic potentials, are harmonic and Eq. (13) can be rearranged to give 1 U  kx 2 ; 2

(20)

k  kphoto  kgradient − kelastic :

(21)

with The minus sign is introduced for kelastic and the  sign is introduced for k gradient because the elastic force opposes, and the optical gradient force may add to or oppose the photoisomerization force, respectively (vide infra). The distribution at equilibrium, which is reached for infinite times, is 1∕2 

k kx 2 : (22) exp − Ψeq
x  2kB T 2πkB T Given that U  12 kx 2 is the potential energy of the harmonic oscillator, the distribution is identified as the thermal distribution of the oscillator [79]

Research Article 
U
x : Ψeq
x ∝ exp − kB T

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(23)

Such an expression is also known as the probability distribution function for the displacement of a trapped particle in optical tweezers at thermodynamic equilibrium [84]. Ψeq
x is a Gaussian with 1∕2
k ; (24) a 2πk B T the height of the curve’s peak, and 
k −1∕2 ; b kB T

(25)

the width of the bell relatedp toffiffiffiffiffiffiffiffiffiffiffiffi the full width at half-maximum (FWHM) by FWHM  2 2 ln 2b. The bottom image in Figure 4 shows an atomic force microscopy (AFM) image and the related line scan of the deformation pattern created by a tightly focused laser beam, considered here as a point source, in an azo-polymer film. Clearly, the shape of the deformation is Gaussian. Also, it was experimentally observed that the shape of the deformation is Gaussian when the light beam is a line or a series of lines obtained from interference patterns [32–37]. More remarkably is the intensity dependence of the height and width of the mass movement. Indeed, it was shown that the height a increases with the increased intensity [42]. In fact, the photoisomerization force is dominant and in most of the experiments performed to date, except a few [30,40–44], only the photoisomerization force is observed. I therefore assume k  kphoto pffiffiffiffiffi and obtain pffiffiffiffiffi from Eqs. (6), (24), and (25) that a ∝ I 0 and b ∝ 1∕ I 0 , indicating the larger the irradiation light intensity, the larger the height of and the sharper the Gaussian-type deformation. It was unambiguously shown that a increases with the increasing intensity, and additional experiments are needed to verify the square-root dependence of a and 1∕b on the intensity at equilibrium. The same type of experiments, e.g., the study of the dependence of a and 1∕b on the intensity at equilibrium, may be conducted to quantify the influence of the polymer’s elasticity on photo migration whereby k elastic is measured, using Eq. (21), for different types of azo polymers.

265

e.g., t ≫ τ, G
x; x 0 ; t is the Boltzmann distribution [Eq. (22)]. At any given time t, the distribution [Eq. (26)] is Gaussian with a variancepgiven ffiffiffiffiffiffiffiffiffi by B
t [Eq. (27)] and a standard deviation given by B
t and a mean given by A
t [Eq. (28)] as B
t 

kB T
1 − exp
−2t∕τ; k

A
t  x 0 exp
−t∕τ:

(27) (28)

Figure 6 shows the distribution at a time near zero, which corresponds to free diffusion, and at infinite times, which correspond to photomigration equilibrium. It also shows the distribution at intermediate times. The height of the distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffi decreases from 1∕ 4πDt at free diffusion to 1∕ 2πk B T ∕k at photomigration equilibrium, and the mean, which is the point at which the probability distribution function is maximum, is shifting exponentially from x 0 at t ∼ 0 to zero at t ∼ ∞ for free diffusion versus equilibrium, respectively. The fact that the maximum of the distribution decreases with time means that less particles are Brownian. Similarly, the distribution is narrower at free diffusion because all the particles present a Brownian motion, and it broadens as time passes due photomigration of the particles. Other direct consequences of Eq. (26) also include the fact that the height and the width and the mean dynamically depend as well on the irradiation intensity I 0, via τ (τ  ξ∕k) and k ∝ I 0 . It was reported that the height and the FWHM of the photoinduced deformation pattern observed on an azo polymer exhibit an exponential dependence on the exposure time and irradiation intensity, e.g., depend on I, and the larger the irradiation intensity, the faster the exponential growth of the height and the FWHM [42]—a feature which is in agreement with the theoretical predictions (τ  ξ∕k). τ can be measured by following the position of the center of the Gaussian, e.g., the position that corresponds to the maximum of the mass distribution [for example, the position marked by the vertical arrow in Fig. 4, bottom, with time and fitting by Eq. (28)]. The diffusion constant D (or the mobility ξ) of the particles with the

B. Dynamics of Photomigration

The dynamics of photomigration can be described by the Green’s function G
x; x 0 ; t which is the probability that the system which was in the state x 0 at time t  0 is in the state x at time t. G
x; x 0 ; t is given by

−1∕2  2πkB T 2t 0 G
x; x ; t  1 − exp − k τ   0 kx − x exp
−t∕τ2 : (26) × exp − 2kB T
1 − exp
−2t∕τ This expression is the distribution function of the onedimensional Brownian motion of a particle in a harmonic potential [79,85]. In our case, the harmonic potential is the photoisomerization one. Note that such an expression may as well describe the dynamics of optical tweezing for a harmonic potential. When t is small, e.g., t ≪ τ, G
x; x 0 ; t is the distribution of free diffusion [Eq. (19)], and when t is large,

Fig. 6. Distribution function of photomigration [Eq. (26)] at times near zero (1), which corresponds to free diffusion, and at infinite times (6 and 7), which correspond to near photomigration equilibrium, and at intermediate times. The curves correspond to values of t∕τ according to the series 002 (1); 005 (2); (3); 0.5 p0.2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (4); 1 (5); 2 (6); 10 (7). G
x; x 0 ; t is normalized by 1∕ 2πk B T ∕k.

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chromophore in the trans state can be measured, at the early time evolution when t ≪ τ, from the slope of the height versus time t; τ and k (τ  ξ∕k) can be measured from equilibrium for various irradiation intensities. Such experiments allow for not only the measurement of the material’s parameters but also must give insight about the different contributions to the observed photomigration, e.g., to k, of the photoisomerization, optical gradient, and elastic forces. I now calculate the displacement λx in the direction of the x-axis, e.g., the direction of the photomigration movement, that a particle experiences on the average, e.g., more precisely, the square root of the arithmetic mean of the squares the pof ffiffiffiffiffiffiffiffi ffi displacements in the direction of the x-axis ( B
t). Equations (17) and (27) and (τ  ξ∕k) show that λx  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi B
t increases from 2Dt , corresponding to free diffusion, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi to kB T ∕k corresponding to photomigration equilibrium. Taking the p upper limit ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi for the photomigration displacement, e.g., λx  kB T ∕k , and using kB T  411 × 10−23 J at T  298 K; and Eq. (6) for k, e.g., k  N δσ T ϕT C I, with N  1021 molecules · cm−3 ; δ  5 × 10−6 cm; ϕT C  0.11 nm and σ 490  11.6 × 10−17 cm2 molecule−1 [86] for DR1 in T PMMA [49]; and considering millijoules of light energy incident on a 400 nm diameter area, which is defined by the size of the laser spot and it corresponds to the case of Fig. 4, I  8 × 10−10 J cm−2 , I find λx  89 nm. Such a value depends on the size of the irradiating spot, and it is consistent with the observed amount of mass photomigration in an azo polymer irradiated in the conditions described above, whereby the mass displaced in 1 s of irradiation is located about 200 nm from the center of the laser irradiation [42]. The magnitude of the force
kλx  which moved the azo polymer by 89 nm is ∼0.05 pN, and it is well within the range of optical tweezing forces (100 fN–100 pN), and the work done by this force
kλ2x  is ∼0.004 fJ. Given the amount of displaced volume of the azo polymer, e.g., ∼40 nm3 estimated for this experiment, and the typical chromophores density given above, the force per molecule is ∼1.25 fN, and the work done per unit volume displaced, in a cubic nanometer, and per azo chromophore is ∼10−4 fJ. It is remarkable that such a small amount of work can lead to yet a small amount, albeit appreciable, photoisomerizationinduced displacement of matter. 4. CONCLUSION Optical tweezing in azo polymers is due to molecular photomigration. The azo polymer’s motion is governed by four competing movements: the photoisomerization movement, and movements due to the restoring optical gradient and elastic forces, as well as the spontaneous diffusion due to Brownian motion. Phenomenologically, the photoisomerization force pushes the molecules from high-intensity to low-intensity regions, and the optical gradient force together with the force due to the elasticity of the polymer pull the molecules back to their initial positions. The photoisomerization force is a kind of optical force, and it requires the conjunction of two effects to occur, e.g., the existence of a gradient of light intensity which is strongly polarization dependent and photoisomerization. The specificity of such a force is that it is a repelling force and that its

direction corresponds to that of light polarization. Let us hope that researchers will use the formalism developed here, which is of such importance in photomigration as well as in optical nanomanipulation as it opens new research perspectives. Funding.

Japan-Morocco Handai project.

REFERENCES AND NOTE 1. J. H. Poynting, “On the transfer of energy in the electromagnetic field,” Philos. Trans. R. Soc. London 175, 343–361 (1884). 2. P. Lebedev, “Untersuchungen über die druckkräfte des lichtes,” Ann. Phys. 311, 433–458 (1901). 3. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901). 4. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970). 5. A. Ashkin, “Atomic-beam deflection by resonance-radiation pressure,” Phys. Rev. Lett. 25, 1321–1324 (1970). 6. S. Chu, “The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998). 7. C. Cohen-Tannoudji, “Manipulating atoms with photons,” Rev. Mod. Phys. 70, 707–719 (1998). 8. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectricparticles,” Opt. Lett. 11, 288–290 (1986). 9. A. Ashkin, “History of optical trapping and manipulation of small neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000). 10. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). 11. O. M. Maragò, P. J. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013). 12. A. Ishijima and T. Yanagida, “Single molecule nanobioscience,” Trends Biochem. Sci. 26, 438–444 (2001). 13. Y. Ishii, A. Ishijima, and T. Yanagida, “Single molecule nanomanipulation of biomolecules,” Trends Biotechnol. 19, 211–216 (2001). 14. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). 15. D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–271 (1997). 16. A. Janshoff and C. Steinem, “Transport across artificial membranesan analytical perspective,” Angew. Chem. Int. Ed. Engl. 39, 3213–3237 (2000). 17. B. Onoa, S. Dumont, J. Liphardt, S. B. Smith, I. Tinoco, and C. Bustamante, “Identifying kinetic barriers to mechanical unfolding of the T. thermophila ribozyme,” Science 299, 1892–1895 (2003). 18. J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, and C. Bustamante, “Reversible unfolding of single RNA molecules by mechanical force,” Science 292, 733–737 (2001). 19. Z. Bryant, M. D. Stone, J. Gore, S. B. Smith, and N. R. Cozzarelli, “Bustamante, C. structural transitions and elasticity from torque measurements on DNA,” Nature 424, 338–341 (2003). 20. S. B. Smith and Y. Cui, “Bustamante, C. overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996). 21. Z. Sekkat and W. Knoll, Photoreactive Organic Thin Films (Academic, 2002). 22. Z. Sekkat and S. Kawata, “Laser nanofabrication in photoresists and azopolymers,” Laser Photon. Rev. 8, 1–26 (2014). 23. A. Priimagi and A. Kravchenko, “Azopolymer-based micro- and nanopatterning for photonic applications,” J. Polym. Sci. 52, 163–182 (2014). 24. R. Won, “Azopolymer option,” Nat. Photonics 5, 649 (2011). 25. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and M. Pasqualino, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. 3, 989 (2012).

Research Article 26. M. Watabe, G. Juman, K. Myamoto, and T. Omatsu, “Light induced conch-shaped relief in an azo-polymer film,” Sci. Rep. 4, 4281 (2014). 10.1038/srep04281. 27. G. J. Fang, J. E. Maclennan, Y. Yi, M. A. Glaser, M. Farrow, E. Korblova, D. M. Walba, T. E. Furtak, and N. A. Clark, “Athermal photofluidization of glasses,” Nat. Commun. 4, 1521 (2013). 28. P. Karageorgiev, D. Neher, B. Schulz, B. Stiller, U. Pietsch, M. Giersig, and L. Brehmer, “From anisotropic photo-fluidity towards nanomanipulation in the optical near-field,” Nat. Mater. 4, 699–703 (2005). 29. S. Lee, H. S. Kang, and J.-K. Park, “Directional photofluidization lithography: micro/nanostructural evolution by photofluidic motions of azobenzene materials,” Adv. Mater. 24, 2069–2103 (2012). 30. H. Ishitobi, I. Nakamura, T. Kobayashi, N. Hayazawa, Z. Sekkat, S. Kawata, and Y. Inouye, “Nanomovement of azo-polymers induced by longitudinal fields,” ACS Photon. 1, 190–197 (2014). 31. S. S. Kharintsev, A. I. Fishman, S. G. Kazarian, I. R. Gabitov, and M. K. Salakhov, “Experimental evidence for axial anisotropy beyond the diffraction limit induced with a bias voltage plasmonic nanoantenna and longitudinal optical near-fields in photoreactive polymer thin films,” ACS Photon. 1, 1025–1032 (2014). 32. A. Kravchenko, A. Shevchenko, V. Ovchinnikov, A. Priimagi, and M. Kaivola, “Optical interference lithography using azobenzenefunctionalized polymers for micro- and nanopatterning of silicon,” Adv. Mater. 23, 4174–4177 (2011). 33. P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995). 34. D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995). 35. S. Bian, L. Li, J. Kumar, D. Y. Kim, J. Williams, and S. Tripathy, “Single laser beam-induced surface deformation on azobenzene polymer films,” Appl. Phys. Lett. 73, 1817–1819 (1998). 36. S. Bian, J. M. Williams, D. Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy, “Photoinduced surface deformations on azobenzene polymer films,” J. Appl. Phys. 86, 4498–4508 (1999). 37. A. Sobolewska and A. Miniewicz, “On the inscription of period and half-period surface relief gratings in azobenzene-functionalized polymers,” J. Phys. Chem. B 112, 4526–4535 (2008). 38. Y. Gilbert, R. Bachelot, P. Royer, A. Bouhelier, P. Wiederrrecht, and L. Novotny, “Longitudinal anisotropy of the photoinduced molecular migration in azobenzene polymer films,” Opt. Lett. 31, 613–615 (2006). 39. X. L. Jiang, L. Li, J. Kumar, D. Y. Kim, and S. K. Tripathy, “Unusual polarization dependent optical erasure of surface relief gratings on azobenzene polymer films,” Appl. Phys. Lett. 72, 2502–2504 (1998). 40. R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J.-P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003). 41. H. Ishitobi, M. Tunabe, Z. Sekkat, and S. Kawata, Nanomovement of azo polymers induced by metal tip enhanced near-field irradiation,” Appl. Phys. Lett. 91, 091911 (2007). 42. H. Ishitobi, M. Tanabe, Z. Sekkat, and S. Kawata, “The anisotropic nanomovement of azo-polymers,” Opt. Express 15, 652–659 (2007). 43. Y. Gilbert, R. Bachelot, A. Vial, G. Lerondel, P. Royer, A. Bouhelier, and G. P. Wiederrecht, “Photoresponsive polymers for topographic simulation of the optical near-field of a nanometer sized gold tip in a highly focused laser beam,” Opt. Express 13, 3619–3624 (2005). 44. F. L. Labarthet, J. L. Bruneel, T. Buffeteau, and C. Sourisseau, “Chromophore orientations upon irradiation in gratings inscribed on azo-dye polymer films: a combined AFM and confocal Raman microscopic study,” J. Phys. Chem. B 108, 6949–6960 (2004). 45. M. Hasegawa, T. Ikawa, M. Tsuchimori, O. Watanabe, and Y. Kawata, “Topographical nanostructure patterning on the surface of a thin film of polyurethane containing azobenzene moiety using the optical near field around polystyrene spheres,” Macromolecules 34, 7471–7476 (2001). 46. H. Rau, “Photoisomerization of benzenes,” in Photoreactive Organic Thin Films (Academic, 2002), Chap. 1, pp. 3–28.

Vol. 55, No. 2 / January 10 2016 / Applied Optics

267

47. Z. Sekkat, “Photo-orientation by photoisomerization,” in Photoreactive Organic Thin Films (Academic, 2002), Chap. 3, pp. 64–102. 48. Z. Sekkat, D. Yasumatsu, and S. Kawata, “Pure photo-orientation of azo dye in polyurethanes and quantification of spectrally overlapping isomers,” J. Phys. Chem. B 106, 12407–12417 (2002). 49. R. Loucif-Saibi, K. Nakatani, J. Delaire, Z. Sekkat, and M. Dumont, “Photoisomerization and second harmonic generation in disperse red one-doped and functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993). 50. Z. Sekkat, J. Wood, and W. Knoll, “Reorientation mechanism of azobenzenes within the trans-cis photoisomerization,” J. Phys. Chem. 99, 17226–17234 (1995). 51. S. Brasselet and J. Zyss, “Multipolar molecules and multipolar fields: probing and controlling the nature of nonlinear molecular media,” J. Opt. Soc. Am. B 15, 257–265 (1998). 52. J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349–354 (1994). 53. J. Kumar, L. Li, X. Jiang, D.-Y. Kim, T. Lee, and S. Tripathy, “Gradient force: the mechanism for surface relief grating formation in azobenzene functionalized polymers,” Appl. Phys. Lett. 72, 2096–2098 (1998). 54. K. Yang, S. Yang, and J. Kumar, “Formation mechanism of surface relief structures on amorphous azopolymer films,” Phys. Rev. B 73, 165204 (2006). 55. C. J. Barrett, P. L. Rochon, and A. L. Natansohn, “Model of laserdriven mass transport in thin films of dye-functionalized polymers,” J. Chem. Phys. 109, 1505–1516 (1998). 56. P. Lefin, C. Fiorini, and J.-M. Nunzi, “Anisotropy of the photo-induced translation diffusion of azobenzene dyes in polymer matrices,” Pure Appl. Opt. 7, 71–82 (1998). 57. B. Bellini, J. Ackermann, H. Klein, C. Grave, P. Dumas, and V. Safarov, “Light-induced molecular motion of azobenzene-containing molecules: a random-walk model,” J. Phys. Condens. Matter 18, S1817–S1835 (2006). 58. T. G. Pedersen, P. M. Johansen, N. C. R. Holme, P. S. Ramanujam, and S. Hvilsted, “Mean-field theory of photoinduced formation of surface reliefs in side-chain azobenzene polymers,” Phys. Rev. Lett. 80, 89–92 (1998). 59. K. Sumaru, T. Yamanaka, T. Fukuda, and H. Matsuda, “Photoinduced surface relief gratings on azopolymer films: analysis by a fluid mechanics model,” Appl. Phys. Lett. 75, 1878–1880 (1999). 60. K. Sumaru, T. Fukuda, T. Kimura, H. Matsuda, and T. Yamanaka, “Photoinduced surface relief formation on azopolymer films: a driving force and formed relief profile,” J. Appl. Phys. 91, 3421–3430 (2002). 61. M. L. Juan, J. Plain, R. Bachelot, P. Royer, S. K. Gray, and G. P. Wiederrecht, “Stochastic model for molecular transport in polymer films,” Appl. Phys. Lett. 93, 153304 (2008). 62. M. L. Juan, J. Plain, R. Bachelot, P. Royer, S. K. Gray, and G. P. Wiederrecht, “Multiscale model for photoinduced molecular motion in azo polymers,” ACS Nano 3, 1573–1579 (2009). 63. H. Galinski, “Instability-induced pattern formation of photoactivated functional polymers,” Proc. Natl. Acad. Sci. U.S.A. 111, 17017–17022 (2014). 64. V. Toshchevikov, V. P. Toshchevikov, M. Saphiannikova, and G. Heinrich, “Microscopic theory of light induced deformation in amorphous side-chain azobenzene polymers,” J. Phys. Chem. B 113, 5032–5045 (2009). 65. J. M. Ilnytskyi, “Opposite photoinduced deformations in azobenzene containing polymers with different molecular architecture: molecular dynamics study,” J. Chem. Phys. 135, 044901 (2011). 66. A. Voit, A. Krekhov, W. Enge, L. Kramer, and W. Kohler, “Thermal patterning of a critical polymer blend,” Phys. Rev. Lett. 94, 214501 (2005). 67. A. Voit, A. Krekhov, and W. Kohler, “Laser induced structures in a polymer blend in the vicinity of the phase boundary,” Phys. Rev. E 76, 011808 (2007). 68. D. Bublitz, B. Fleck, and L. Wenke, “A model for surface-relief formation in azobenzene polymers,” Appl. Phys. B 72, 931–936 (2001).

268

Research Article

Vol. 55, No. 2 / January 10 2016 / Applied Optics

69. Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (1992). 70. Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dye in polymeric films. Characterization of molecular angular mobility,” Synth. Met. 54, 373–381 (1993). 71. Z. Sekkat and W. Knoll, “Creation of second order nonlinear optical effects by photoisomerization of polar azo dyes in polymeric films: theoretical study of steady-state and transient properties,” J. Opt. Soc. Am. B 12, 1855–1867 (1995). 72. Z. Sekkat, “Two-photon assisted poling,” J. Opt. Soc. Am. B 27, 132–140 (2010). 73. Z. Sekkat, J. Wood, W. Knoll, W. Volksen, and R. D. Miller, “Lightinduced orientation in a high glass transition temperature polyimide with polar azo dyes in the side chain,” J. Opt. Soc. Am. B 13, 1713–1724 (1996). 74. Z. Sekkat, P. Pretre, A. Knoesen, W. Volksen, V. Y. Lee, R. D. Miller, J. Wood, and W. Knoll, “Correlation between polymer architecture and sub-glass-transition-temperature light-induced molecular movement in azo-polyimide polymers: influence on linear and second- and third-order nonlinear optical processes,” J. Opt. Soc. Am. B 15, 401–413 (1998). 75. S. Serak, N. Tabiryan, T. J. White, R. A. Vaia, and T. J. Bunning, “Photogeneration of work from polymers,” Soft Matter 6, 779–783 (2010). 76. J. P. Abid, M. Frigoli, R. Pansu, J. Szeftel, J. Zyss, C. Larpent, and S. Brasselet, “Light-driven directed motion of azobenzene-coated

77. 78. 79. 80.

81. 82.

83.

84.

85. 86.

polymer nanoparticles in an aqueous medium,” Langmuir 27, 7967–7971 (2011). J. Durnin, “Exact Solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). T. Grosjean and D. Courjon, “Photopolymers as vectorial sensors of the electric field,” Opt. Express 14, 2203–2210 (2006). M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, 1994). A. Einstein, “Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen,” Ann. Phys. 17, 549–568 (1905). G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University, 1970), Chap. 4. T. White, S. V. Serak, N. V. Tabiryan, R. A. Vaia, and T. J. Bunning, “Polarization-controlled, photodriven bending in monodomain liquid crystal elastomer cantilevers,” J. Mater. Chem. 19, 1080–1085 (2009). T. Ube and T. Ikeda,”Photomobile polymer materials with crosslinked liquid-crystalline structures: molecular design, fabrication, and functions,” Angew. Chem. Int. Ed. Engl. 53, 10290–10299 (2014). E. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998). G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Phys. Rev. 36, 823–841 (1930). nm nm nm nm σ 490 is calculated from σ 490  3.82 × 10−21 ε490 , where ε490  T T T T 4 −1 −1 3.04 × 10 L ⋅ mol cm is the isotropic extinction coefficient of transDR1 in a PMMA copolymer [29].