Photoinduced surface relief formation on azopolymer ...

Photoinduced surface relief formation on azopolymer films: A driving force and formed relief profile Kimio Sumaru, Takashi Fukuda, Tatsumi Kimura, Hiro Matsuda, and Tadae Yamanaka Citation: Journal of Applied Physics 91, 3421 (2002); doi: 10.1063/1.1432482 View online: http://dx.doi.org/10.1063/1.1432482 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Alteration of the mechanical properties of azopolymer film in the process of surface relief grating formation Appl. Phys. Lett. 94, 231911 (2009); 10.1063/1.3153847 Efficient formation of surface relief grating on azopolymer films by gold nanoparticles J. Appl. Phys. 104, 103117 (2008); 10.1063/1.3031278 Confinement of surface patterning in azo-polymer thin films J. Chem. Phys. 126, 094908 (2007); 10.1063/1.2538787 Relaxation kinetics of photoinduced surface relief grating on azopolymer films J. Appl. Phys. 92, 6959 (2002); 10.1063/1.1516264 Photoinduced surface relief gratings on azopolymer films: Analysis by a fluid mechanics model Appl. Phys. Lett. 75, 1878 (1999); 10.1063/1.124858

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 150.29.238.151 On: Tue, 03 Feb 2015 07:24:58

JOURNAL OF APPLIED PHYSICS

VOLUME 91, NUMBER 5

1 MARCH 2002

Photoinduced surface relief formation on azopolymer films: A driving force and formed relief profile Kimio Sumarua) Materials System Group, Institute for Materials and Chemical Process, National Institute of Advanced Industrial Science and Technology, Tsukuba Central 5, 1-1-1, Higashi, Tsukuba, Ibaraki 305-8565, Japan

Takashi Fukuda, Tatsumi Kimura, and Hiro Matsuda Molecular Photonics Group, Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba Central 5, 1-1-1, Higashi, Tsukuba, Ibaraki 305-8565, Japan

Tadae Yamanaka Materials System Group, Institute for Materials and Chemical Process, National Institute of Advanced Industrial Science and Technology, Tsukuba Central 5, 1-1-1, Higashi, Tsukuba, Ibaraki 305-8565, Japan

共Received 31 July 2001; accepted for publication 12 November 2001兲 A driving force of photoinduced surface relief 共PSR兲 formation is examined by considering the complex electric susceptibility of the absorbing medium. It is introduced into a fluid mechanics model established for the viscous fluid layer on a flat substrate. As a result, a theoretical model for PSR formation covering thoroughly from the origin of the driving force to the dynamic process forming the surface relief is established. A series of systematic experiments are carried out by means of one-dimensional sinusoidal and Gaussian beam irradiation and the results are reproduced well by the model calculation. The experimental results reported in former literature are also reviewed using this model and the result supports strongly our assumption: driving force derived from the interaction between the absorbing medium and optical electric field acts as a body force on the optically plasticized azopolymer. Further, from the consideration based on the model, some important features of PSR formation which have not been reported yet are suggested. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1432482兴 I. INTRODUCTION

In 1995, it was discovered that thin films of an azobenzene containing polymer 共azopolymer兲 form a surface modulation under the irradiation with an interference pattern of a laser beam; this phenomenon is now known as a photoinduced surface relief 共PSR兲 formation.1,2 Since the relief pattern can be inscribed in a single-step process of light irradiation, its application to the field of optical data storage and nanofabrication is strongly anticipated.3 Aiming at the clarification of its whole mechanism and the establishment of its application technology, PSR formation has been studied actively in both experimental and theoretical manners by many research groups in the world. As the first attempt to investigate the mass transporting properties in this phenomenon, Kim et al. demonstrated the orthogonal inscription of two sets of PSR gratings.2 On a set of PSR gratings of an azopolymer thin film, interfering two laser beams were irradiated to append another orthogonal set of gratings. As a result, appended gratings were superposed on initial gratings and the direction of the mass transfer in the PSR grating formation was turned to be restricted along its wave number vector. Jiang et al. examined the PSR grating formation in various polarization conditions of irradiating light for the first time4 and later, Viswanathan et al. carried out a series of systematic investigation.5 They showed that this phenomenon has a very strong dependence on light polarization, and especially that a component of the optical a兲

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electric field along the wave number vector of gratings is essential to the mass transfer. Besides the periodic light intensity, Kumar et al. used the intensity distribution produced by the edge diffraction.6 Furthermore, Bian et al. used a single focused laser beam to realize the condition of local irradiation, and made it clear that the mass transfer in the PSR formation occurs from the bright region to the dark region.7 Many experimental results were reported also on the dynamic processes of the formation and erasure of PSR gratings. Viswanathan et al. showed that the PSR grating formation is suppressed when the transparent multilayer of the polyion complex is coated on the azopolymer surface.8 Bian et al. and Barret et al. obtained the linear relationship between the rate of the PSR formation and light intensity.9,10 Fukuda et al. carried out the experiment using amorphous azopolymers and investigated quantitatively the dependence of the PSR formation on the azofunctionalization,11,19 the temperature11 and the intensity of the irradiating light.12,19 As to the optical erasure of the PSR, Jiang et al. carried out a series of systematic experiments; PSR gratings, which had been prepared under the several conditions of light polarization in advance, were irradiated with the uniform beam in several polarization conditions.13 They reported that the PSR was kept stable under the irradiation of a uniform beam when a certain combination of light polarization for inscribing and erasing processes was chosen, and also that it was even enhanced on another condition. All these are important evidence to outline the whole mechanism of PSR formation.

0021-8979/2002/91(5)/3421/10/$19.00 3421 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 150.29.238.151 On: Tue, 03 Feb 2015 07:24:58

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J. Appl. Phys., Vol. 91, No. 5, 1 March 2002

Corresponding to these experimental results, several models were proposed to describe an elementary process providing a driving force of PSR formation. Barrett et al. considered that the distribution of internal pressure induced by the volume change of photoisomerized azobenzene brings about mass transport.10,14 Although this idea could not explain the effect of the polarization condition on spinning rotor guage 共PSR兲 formation, it provided a new picture to the dynamic formation of PSR: mass transport due to the bulk deformation of viscous fluid. Pedersen et al. paid attention to the interaction among dipoles of azobenzenes to account for the general dependence on the polarization condition of irradiating light.15 Based on the minimizing condition of an energy functional concerning an influence of surface tension, they formulated the undulation of PSR in an equilibrium state. However, their resultant formula contradicts to the direction of mass transport which is generally observed for azopolymer thin layers.7 Lefin considered an anisotropic diffusion arisen from the trans-cis-trans cycling photoisomerization of azobenzene chromophores, but the polarization dependence deduced in this model is much smaller than that observed experimentally.16 Kumar attributed the PSR driving force to an optically induced gradient force and derived a force which is proportional to the first order derivative of light intensity with respect to the position in the film plane.6,17 For a mechanism supposed in this theory to meet the experimentally observed direction of mass transport, there is still much left for further examination. However, they succeeded to explain reasonably the general polarization dependence of PSR formation. Besides the elementary process providing a driving force, there have been many discussions on the dynamics of PSR formation and the resultant relief profile. Barrett et al. considered the mass transport in PSR formation as the bulk deformation of viscous fluid.10 Solving a simplified Navier– Stokes equation, they succeeded to explain the film thickness 共h兲 dependence observed experimentally in the range of small h value. In their model, however, velocity distribution in the film surface was ignored and driving force was assumed to be independent of the position in the depth direction. The theory thus could not account for the film thickness dependence in the range of large h. Based on the incompressibility of the azopolymer, Bian et al. showed that the resultant PSR profile is proportional to the first order derivative of the driving force with respect to the position in the film plane when the force and fluid velocity have no depth dependence.9 Introducing an optically induced gradient force proposed by Kumar et al.,6 this model suggests that the PSR profile is proportional to the second order derivative of light intensity, and can provide the qualitative explanation to the experimental result with a Gaussian single beam. In this situation, we proposed a fluid mechanics model, which takes depth dependence of the PSR driving force and velocity distribution in the film surface into account.18 Twodimensional distribution of the mass transfer velocity field was solved for a Navier–Stokes equation considering the influence of the substrate as a fixed boundary and large optical density of the azopolymer film. In the model setting, the PSR driving force was assumed as a sinusoidal body force

with exponentially decaying distribution in the depth direction, and no physical elementary process was specified for the generation of force. The rate of the PSR formation was obtained as a function of film thickness and interference wave number, and agreed well with the experimental data collected in our diffraction efficiency measurement. Although a reasonable model was thus established in terms of the dynamics of PSR formation, no feasible mechanism with sufficient physical background has ever been proposed for the elementary process providing the PSR driving force, which is the core of the problem, and the entire picture of this phenomenon has been still left unclear. Our study reported here is to examine several experimental features of the PSR formation collectively and to construct a fundamental theory revealing its whole mechanisms. In the next section, we consider the interaction between the absorbing medium and optical electric field in detail and deduce a formula with respect to the PSR driving force, which is consistent with most experimental results. Then this formulation is applied into a fluid mechanics model in terms of the viscous thin layer on a substrate, a theory covering thoroughly from the origin of the driving force to the dynamic process forming PSR is established. In Sec. III, the systematic data of the PSR collected for one dimensional 共1D兲 sinusoidal and 1D Gaussian intensity distributions of irradiating light are reported and these data are examined quantitatively by our theory. Additionally to the result of the examination in Sec. III, other important features of the PSR are discussed on the basis of our model, and an availability of the model as a theoretical foundation in considering the PSR formation is investigated in detail.

II. THEORY A. Driving force of PSR formation

Many experimental results showed that the PSR formation is much dependent on the polarization condition of irradiating light and the possibility of a simple thermal process was ruled out for the elementary process providing a PSR driving force. As reported by Jiang et al. and Viswanathan et al., this polarization dependence is very strong and is up to two orders of magnitude.4,5 In several models proposed till now, the most reasonable one to explain this experimental feature is considered to be a model proposed by Kumar et al. They derived a gradient force on the basis of an interaction between the induced dipoles and the optical electric field.6 A formula for the force vector F deduced by them is described with the optical electric field E at position r and time t as follows: F⫽ 具共 ⑀ 0 ␹ E共 r,t 兲 •ⵜ 兲 E共 r,t 兲典 ⫽ 12 ⑀ 0 ␹ ⬘ E共 r兲 •ⵜE共 r兲 ,

共1兲

where 具典 represents the time average, and ⑀ 0 , ␹, and ␹ ⬘ denote the dielectric constant of vacuum, the complex electric susceptibility of the system and its real part, respectively. On the condition that light intensity is dependent on the x coordinate 共along the film plane兲 alone, the x component of the driving force F x is expressed as


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⳵ I 共 r兲 1 , F x⫽ ⑀ 0␹ ⬘ 4 ⳵x

共2兲

I 共 r兲 ⬅ 兩 E x 共 r兲兩 ,

共3兲

2

9

where E x (r) is the x-component of E(r). In order that the force expressed by Eq. 共1兲 agrees with the experimentally observed direction of mass transfer, ␹ ⬘ should be negative. Kumar et al. and Bian et al. provided a qualitative explanation to this point; the refractive index of the azopolymer could be decreased locally by light irradiation due to the photoisomerization and the reorientation of azochromophores, and the reduction of the refractive index corresponds to negative ␹ ⬘ . 6,9 However, this mechanism implies large dependence of ␹ ⬘ on light intensity I since the decrease of refractive index and the size of the domain with a smaller refractive index should be significantly influenced by I. Due to the multiplication effect of E and ␹ ⬘ , the rate of the PSR formation based on this mechanism should provide much larger I dependence than that observed experimentally, and the PSR profile should be much changed by I. In order to settle this inconsistency with experimental results, we draw our attention to the large absorption coefficient of the azopolymer and consider the interaction between the absorbing medium and optical electric field. At first, we consider the microscopic structure of the azopolymer. It is reasonable to provide the azopolymer with a microscopic picture in which azochromophores are dispersed in a transparent matrix as absorbing bodies. This is a good expression for the microscopic heterogeneity of the practical system implying that the absorbing nature of an azopolymer is brought mostly by a dispersed azochromophore and has a greatly localized distribution. To describe this situation concisely, an effective volume fraction of the azochromophore ␾ is introduced here. This can be considered as a proportion of the absorbing region in the azopolymer and its smaller value implies the greater localization of the absorbing nature. Although this value should be proportional to the dye functionalization, it is not necessarily corresponding to the practical volume fraction. The absorption coefficient of the azopolymer ␣, which is defined overall, is related to that in absorbing region ␣ dye by the following equation:

␣ ⫽ ␣ dye␾ .

共4兲

Subsequently, we examine the dielectric properties of the azopolymer concerning this microscopic heterogeneity of the azopolymer. The real part of susceptibility of the ith phase composing azopolymer 共i⫽‘‘dye’’ or ‘‘matrix’’兲 is described as

␹ ⬘i ⫽n 2i ⫺

␣ 2i ⫺1, 4K 2

and of the transparent matrix, which are denoted respectively ⬘ and ␹ matrix ⬘ , are described as follows: by ␹ dye 2 ⬘ ⫽n dye ␹ dye ⫺

using the following definition:

共5兲

where K, n i , and ␣ i are the wave number of irradiating light in vacuo, the refractive index, and the absorption coefficient of the ith phase, respectively 共see Appendix兲. Accordingly, the real parts of the susceptibility of the absorbing dye region

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2 ␣ dye ␣2 2 ⫺1⫽n ⫺ ⫺1, dye 4K 2 4K 2 ␾ 2

共6兲

2 ⬘ ⫽n matrix ␹ matrix ⫺1,

共7兲

where n dye and n matrix are the refractive indices of each region, respectively. Since the body force is additive, the gradient force acting on the azopolymer is described finally as F x⫽ ␤

⳵ I 共 r兲 , ⳵x

共8兲

where

␤⬅ ⫽

⑀0 ⬘ ⫹ 共 1⫺ ␾ 兲 ␹ matrix ⬘ 兴关 ␾ ␹ dye 4

冋

册

⑀0 2 ␣2 2 2 n matrix⫹ ␾ 共 n dye ⫺n matrix ⫺1 . 兲⫺ 4 4K 2 ␾

共9兲

Although ␤ contains the weighted average of ␹ ⬘ , that is not concerned in the overall susceptibility of the azopolymer 共susceptibility is not an additive in general兲. Basically ␤ is independent of light intensity I and is consistent with the experimental result that the rate of the PSR formation is proportional to I. Furthermore, ␤ described by this equation can be negative in the practical range of ␣, which is observed in the usual azopolymer. Substituting n dye⫽1.7, n matrix ⫽1.5, ␣ ⫽12 ␮ m⫺1 , K⫽12 ␮ m⫺1 and ␾ ⫽0.1 as realistic values, ␤ takes value ⬃⫺0.3⑀ 0 to provide a force from the higher intensity region to the lower region, and is consistent with the experimentally observed direction of mass transfer. Further, ␤ takes a finite positive value for a completely transparent medium ( ␣ ⫽0) and suggests an important possibility that the PSR can be formed even on the surface of transparent materials.

B. Navier–stokes equation

In order to establish a model to describe the dynamic process of the PSR formation, here we consider the relation between the acting body force and resultant dynamics of the medium. At first, taking many experimental results hitherto reported into account, we assume that the azopolymer behaves as a viscous fluid in the PSR forming process and the mass transfer occurs as a result of bulk deformation brought by the driving force. Although some experimental results suggest the existence of threshold intensity as to PSR formation, the viscosity of the medium is assumed to be uniform and constant independent of local light intensity and the position for simplicity here. This assumption can be set in correspondence to the practical situation that the intensity of bias light is large enough. Also it is consistent with the experimental results that the rate of PSR formation is proportional to light intensity I within the moderate range9,13 and the PSR profile is not changed much by I in a relatively wide range of I. In the experiment of SRG formation, an azopolymer film, which had been prepared by spin coating on a solid


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substrate, is irradiated with laser light from its surface side. This experimental geometry provides the boundary conditions of the fixed end at the interface between polymer and substrate, and that of the free end at the surface of the film. Furthermore, in most practical cases, an optically induced driving force is largely dependent on the depth 共y coordinate兲 since the optical density of the azopolymer film is usually larger than 1 at the wavelength of the irradiating light. In order to settle this asymmetric situation, here we consider a fluid mechanics model in which both the exponential distribution of the optically induced driving force in the y direction and the velocity distribution in the x direction are taken into account. As Barrett et al. discussed, the inertial term and the y component of velocity can be neglected in the Navier– Stokes equation where the viscosity of the polymer is large enough and the PSR undulation is small enough 共laminar flow condition兲.10 Therefore, the simplified Navier–Stokes equation describing the dynamics of the system can be expressed as follows: 0⫽

⳵ 2␯ x ⳵ 2␯ x 1 ⫹ ⫹ F 共 x,y 兲 , ⳵x2 ⳵y2 ␮ x

共10兲

where ␯ x , ␮, and F x (x,y) are the x component of the velocity of mass transport, the viscosity and the body force acting on the polymer as a viscous fluid, respectively. Although Barrett et al. ignored the first term in the right side of Eq. 共10兲 also, it has a slight physical background.10 Supposing that surface modulation is small in the early stage of SRG formation, an opposing surface tension, which Pedersen et al. took into account, is neglected here.10 In the following subsection, Eq. 共10兲 is solved analytically with respect to the 1D sinusoidal distribution of irradiating light intensity on the basis of several earlier assumptions.

FIG. 1. Schematic illustration of our model with involved geometric conditions.

␯ x 共 x,y 兲 ⫽⫺

where C 1 and C 2 are arbitrary constants. From the condition of incompressibility

⳵␯ x ⳵␯ y ⫹ ⫽0, ⳵x ⳵y

I sin共 x,y 兲 ⫽I 0 e ␣ 共 y⫺h 兲共 cos kx⫹1 兲 /2,

␯ y 共 x,y 兲 ⫽⫺ ⫽

where I 0 and k denote the maximum intensity and interference wave number corresponding to the pitch of interference fringe, respectively. The y coordinate is so defined that y ⫽0 at the interface between polymer and substrate and y ⫽h at the surface of polymer. Using Eq. 共8兲, the PSR driving force generated by this intensity distribution of irradiating light F x,sin(x,y) is described as

⳵ 1 F x,sin共 x,y 兲 ⫽ I 0 ␤ e ␣ 共 y⫺h 兲共 cos kx⫹1 兲 2 ⳵x 1 ⫽⫺ I 0 ␤ ke ␣ 共 y⫺h 兲 sin kx. 2

共12兲

On the condition that a force distribution described as Eq. 共12兲, Eq. 共10兲 has a general solution as follows:

冕 ⳵␯⳵

x

x

dy

冋

I 0 ␤ k 2 cos kx 1 ␣ 共 y⫺h 兲 C 1 ky e ⫹ e 2 ␮共 k 2⫺ ␣ 2 兲 ␣ k ⫺

共11兲

共14兲

the y component of the velocity ␯ y is derived as

C. 1D sinusoidal distribution of light intensity

Light intensity in the azopolymer film I sin brought by two-beam interference is expressed by the following equation since it decays exponentially in the y direction as discussed earlier:

I 0 ␤ k sin kx ␣ 共 y⫺h 兲 ⫹C 1 e ky ⫹C 2 e ⫺ky 兲 , 共e 2 ␮共 k 2⫺ ␣ 2 兲共13兲

册

C 2 ⫺ky e ⫹C 3 , k

共15兲

where C 3 is an integral constant. From the boundary conditions of the fixed end at the surface of the substrate ( ␯ x 兩 y⫽0 ⫽0,␯ y 兩 y⫽0 ⫽0), and that of the free end at the surface of the film ( ⳵␯ x / ⳵ y 兩 y⫽h ⫽0), all the arbitrary constants are determined as follows: C 1⫽

⫺ ␣ ⫺ke ⫺ 共 ␣ ⫹k 兲 h , k 共 e ⫺kh ⫹e kh 兲

共16兲

C 2⫽

␣ ⫺ke ⫺ 共 ␣ ⫺k 兲 h , k 共 e ⫺kh ⫹e kh 兲

共17兲

C 3 ⫽⫺

1 ⫺ ␣ h 2 ␣ ⫹ke ⫺ ␣ h 共 e ⫺kh ⫺e kh 兲 e ⫹ . ␣ k 2 共 e ⫺kh ⫹e kh 兲

共18兲

Figure 1 illustrates this situation schematically. Substituting Eqs. 共16兲–共18兲 into Eq. 共15兲, the rate of the PSR formation for the sinusoidal distribution of light intensity R sin is obtained finally in the following equation: R sin共 x 兲 ⬅ ␯ y 兩 y⫽h ⫽I 0 ␤ f 共 k,h, ␣ 兲 cos kx/2␮ ,

共19兲

using a function defined as follows:


Sumaru et al.


f 共 k, ␣ ,h 兲 ⬅

再

冋

册

k2 1 ␣ 1 ⫺ ␣ h 2 ␣ ⫹ke ⫺ah 共 e ⫺kh ⫺e kh 兲 ⫺ ⫺ e ⫹ , 2 k 2 共 e ⫺kh ⫹e kh 兲共k ⫺␣ 兲 ␣ k2 ␣ 2

1 e ⫺2 ␣ h ⫹3⫹2 ␣ h ⫺ , ␣ 2 ␣ 共 e ⫺ah ⫹e ␣ h 兲

lim f 共 k, ␣ ,h 兲 ⫽k 2 h 3 /3,

共21兲

h→0

lim f 共 k, ␣ ,h 兲 ⫽1/␣ .

共22兲

In our previous report, a scaling rule was derived for the function f with a nondimensional film thickness and interference wave number, suggesting a guidance for the treatment of the experimental results obtained with the different samples.18 In the next section, these formulations are generalized for arbitrary 1D distribution of light intensity on the basis of the linearity of the theory.

D. Generalization for arbitrary 1D distribution of light intensity

Here we consider the case that the distribution of light intensity I in the x direction is expressed by an even function P. Using an inverse Fourier transformation Q of the function P,I can be described as follows: I 共 x,y 兲 ⫽I 0 e ␣ 共 y⫺h 兲 P 共 x 兲 2 I e ␣ 共 y⫺h 兲 ␲ 0

冕

⬁

0

Q 共 k 兲 cos kxdk.

共23兲

Thus, the force distribution F x brought by the intensity distribution earlier can be described by using Eqs. 共8兲 and 共12兲 as follows:

⳵ F x 共 x,y 兲 ⫽I 0 ␤ e ␣ 共 y⫺h 兲 P 共 x 兲 ⳵x

冑 ␤ 冑冕

⫽⫺ ⫽2

2 I e ␣ 共 y⫺h 兲 ␲ 0 ⬁

2 ␲

0

冕

lim R 共 x 兲 ⫽ h→0

冑冕冑 ␮␤ 冕

⫽

2 ␲

2 I0 ␲

⬁

0

⬁

0

⫽⫺ lim R 共 x 兲 ⫽ h→⬁

冕

2 I 0h 3␤ ␲ 3␮

⬁

0

Q 共 k 兲 k 2 cos kxdk

I 0h 3␤ ⳵ 2 P共 x 兲 , 3␮ ⳵x2

冑

2 I 0␤ ␲ ␮␣

冕

⬁

0

共26兲

Q 共 k 兲 cos kxdk⫽

I 0␤ P共 x 兲 . ␮␣ 共27兲

Consequently, in the limit of small h, our model suggests that the PSR profile is proportional to the second order derivative of light intensity similarly to Bian’s theory.9 On the other hand, the possibility that the PSR profile coincides with light intensity distribution suggested by Eq. 共27兲 has a significant meaning in applying this phenomenon into nanofabrication. Although the light intensity whose distribution is characterized by an even function is treated here for simplicity, it can be immediately generalized for arbitrary distribution by appending the term of Fourier sine transformation. Next, as an example of generalization, we consider the case that the light intensity has a 1D Gaussian distribution. It is significant to formulate this case since the distribution function is realized with a relatively simple setup of the experiment. A profile function of light intensity which has a 1D Gaussian distribution ( P Gauss) is expressed with Gaussian width ␴ as P Gauss共 x 兲 ⫽e ⫺x

2 /2␴ 2

0

Q Gauss共 k 兲 ⫽ ␴ e ⫺k

kQ 共 k 兲 sin kxdk

Q 共 k 兲 F x,sin共 x,y 兲 dk.

共24兲

Q 共 k 兲 R sin共 k 兲 dk ⬁

冑

共28兲

.

Its inverse Fourier transformation Q Gauss has an analytical expression as follows:

Based on the linearity of the theory, the similar relationship as found in F x between the general function and its Fourier components should be established also in the rate of PSR formation R. Therefore, R corresponding to an arbitrary even function P is expressed by the following equation: R 共 x 兲 ⫽2

共20兲

.

Using Eqs. 共21兲 and 共22兲, the limiting behavior of R is deduced from the general formulas of Fourier transformation as

h→⬁

冑

where k⫽ ␣ ,

where k⫽ ␣

Additionally, the polynomial behavior of the function f in the limit of h→0 and h→⬁ are expressed, respectively, as

⫽

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Q 共 k 兲 f 共 k,h, ␣ 兲 cos kxdk.

共25兲

2 ␴ 2 /2

共29兲

.

Accordingly, the rate of the PSR formation under the irradiation of a 1D Gaussian beam R Gauss is described by the following equation: R Gauss共 x 兲 ⫽

冑

2 I 0␤ ␲ ␮

冕␴ ⬁

0

e ⫺k

2 ␴ 2 /2

f 共 k,h, ␣ 兲 cos kxdk. 共30兲

As defined by Eq. 共20兲, the function f is relatively complicated and it is impossible to obtain an analytical expression for the integration in the right side of Eq. 共30兲. However, Q Gauss has an analytical expression and it approaches 0 rapidly as k increases. Therefore, numerical integration over 关0, 10/␴兴 provides a sufficiently accurate value and it is executable even by using the usual commercial program for nu-


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FIG. 2. Sinusoidal PSR undulation as a function of film thickness. The experiment was carried out on condition of I dosage 14 J/cm2, grating pitch 1.0 ␮m, and absorption coefficient 12.3 ␮m⫺1. Closed circles and solid line indicate the experimental data and corresponding theoretical curve, respectively.

merical calculation. From Eqs. 共26兲 and 共27兲, the limiting behavior of R Gauss is expressed by the very simple equations as follows: lim R Gauss共 x 兲 ⫽⫺ h→0

lim R Gauss共 x 兲 ⫽ h→⬁

I 0h 3␤ 2 2 2 共 x ⫺ ␴ 2 兲 e ⫺x /2␴ , 3␮␴4

I 0 ␤ ⫺x 2 /2␴ 2 e . ␮␣

共31兲

共32兲

III. EXPERIMENTAL RESULTS AND MODEL ANALYSIS A. Materials and apparatus

Although both 1D sinusoidal distribution and 1D Gaussian distribution of irradiating light are available with a relatively simple setup of experiment, they are quite different from each other; one is a periodic function while the other is a localized and aperiodic function. Accordingly, the validity of our model is investigated rationally by the analysis of experimental results obtained for these different distributions. Then we carried out a series of the systematic experiments of PSR formation using a 1D sinusoidal beam and 1D Gaussian beam, and measured the PSR profiles by means of an atomic force microscope 共Nanoscope IIIa, Digital Instruments Co.兲. In the experiment of PSR formation, a 488 nm Ar⫹ laser beam was irradiate on a spin coated film of poly关methylmethacrylate-co-4-关关N-共2-methacryloyloxy兲ethyl兴-N-ethylamino兴-4 ⬘ -nitroazobenzene兴 with 36 mol % dye functionalization 共PMD36兲 on a flat glass substrate. Intensity of irradiating light is higher than the threshold of the PSR formation but low enough to avoid laser ablation. To realize a sinusoidally distributed beam, two circularly polarized flat beams of equal intensity and opposite handedness were irradiated with a certain angle to provide required grating pitch. In order to obtain a 1D Gaussian beam, a two-dimensional 共2D兲 Gaussian beam from a laser

FIG. 3. Sinusoidal PSR undulation as a function of grating pitch. The experiment was carried out on condition of I dosage 7 J/cm2, film thickness 0.8 ␮m, and absorption coefficient 12.3 ␮m⫺1. Closed circles and solid line indicate the experimental data and corresponding theoretical curve, respectively.

light source was focused only in the x direction by using a cylindrical lens. The polarization of the beam was linear along the x direction. B. 1D sinusoidal disutribution

In Fig. 2, the experimental result of a sinusoidal PSR grating formation which is dependent on film thickness h is compared with the corresponding theoretical value. The experiment was carried out on condition of I dosage 14 J/cm2 and grating pitch 1.0 ␮m, and variables ␣, k, and h determined from the experimental condition are used in the calculation of the theoretical curve. The PSR grating amplitude increases sigmoidally with increasing h and approaches a certain value gradually, showing a good agreement with the model. In the small h region, h 3 dependence, which is described well with Eq. 共13兲, is observed in the experimental results obtained by us and Barrett et al.10,18,19

FIG. 4. Log–log plot of the function f vs k/ ␣ . Each f value calculated on several ␣ h conditions is multiplied by the different constant for apparent comparison of the tendency. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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Sumaru et al.


FIG. 5. PSR profiles brought by the irradiation of a 1D Gaussian beam with several values of Gaussian width ␴. The experiment was carried out using azopolymer films of film thickness 1.1 ␮m and absorption coefficient 12.3 ␮m⫺1. Solid lines and dashed lines indicate experimental data and corresponding theoretical curves, respectively.

Figure 3 shows the grating pitch dependence of the PSR grating amplitude formed on a film of thickness 0.8 ␮m with I dosage 7 J/cm2. A maximum is observed in the experimental value around pitch⫽1.0 ␮ m. Although agreement of experimental values with a theoretical curve is not sufficient from the quantitative point of view, also the theoretical curve has a maximum suggesting a feasibility of the model. A similar tendency with a maximum in the grating pitch dependence of the PSR grating amplitude has been reported by also other research groups.10,20 Figure 4 shows the dependence of the function f defined by Eq. 共20兲 on the interference wave number k in a log–log plot. For an apparent comparison of the tendency, each f value on several conditions of ␣ h is multiplied by the different constant respectively. In the limit of ␣ h→0, f is represented by a straight line with slope 2 indicating that the rate of the PSR grating formation is proportional to the second order derivative of light intensity. At larger ␣ h, the asymptotic tendency of f becomes greater suggesting that f shows a weaker k dependence than k 2 in experimentally realizable conditions. In fact, Kumar et al. reported that the PSR profile brought by the edge diffraction is apparently reproduced by the first order derivative of light intensity.6 The tendency of f shown in Fig. 4 can explain their result qualitatively. C. 1D Gaussian distribution

PSR profiles brought by the irradiation of a 1D Gaussian beam with several values of Gaussian width ␴ are compared with corresponding theoretical curves in Fig. 5 (h ⫽1.1 ␮ m, ␣ ⫽12.3 ␮ m⫺1 ). For all values of Gaussian width ␴ from 1.7 to 2.4 ␮m, the theoretical curves agree well with the experimental results. In the range of ␴, the PSR profiles can be reproduced also by Eq. 共31兲 or Bian’s theory.9 Figure 6 shows R Gauss derived as Eqs. 共30兲–共32兲 on several condi-

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FIG. 6. Theoretical R Gauss profiles on several conditions of film thickness. For the model calculation, the values of ␣ and ␴ are set to be 10 ␮m⫺1 and 2 ␮m, respectively.

tions of film thickness. For all theoretical curves in this figure, ␣ and ␴ are set to be 10 ␮m⫺1 and 2 ␮m, respectively, both of which are easily available experimentally, and the depths of the center depression are renormalized to 1. In the limit of h→0, the PSR profile is proportional to the second order derivative of light intensity and the profile calculated for h⫽1 ␮ m is similar to this limiting value. For h ⫽5 ␮ m, two peaks at each side of the central depression are reduced and the profile approaches the limiting value of h →⬁, whose profile is proportional to the distribution of light intensity. Figure 7共a兲 shows the ␴ dependence of the PSR inscription depth brought about on an azopolymer film (h ⫽1.1 ␮ m, ␣ ⫽12.3 ␮ m⫺1 ) by the irradiation of a 1D Gaussian beam (570 J/cm2 @x⫽0). As ␴ increases, the experimental value of the PSR inscription depth decreases rapidly and agrees quite well with the theoretical curve. In Fig. 7共b兲, they are also shown in a log–log plot. Comparing with the dashed line indicating a slope ⫺1.5, it is clearly shown that the experimental data has a ␴ dependence which is intermediate between those of Eqs. 共31兲 and 共32兲. IV. DISCUSSION

In the formula to describe the driving force of the PSR formation, we introduced a coefficient ␤ as shown in Eq. 共9兲, which is corresponding to ⑀ 0 ␹ ⬘ /4 in Kumar’s theory. Since ␤ is indenpendent of the light intensity and can take a negative value when the absorption of the system is large enough, the photoinduced body force derived in our model is consistent with many experimental results as to the intensity dependence and to the direction of mass transfer in the PSR formation. However, some experimental results suggest that ␤ may be changed by the light intensity in the practical cases;


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FIG. 7. ␴ dependence of PSR inscription depth brought about on an azopolymer film 共h⫽1.1 ␮ m, ␣ ⫽12.3 ␮ m⫺1 兲 by the irradiation of a 1D Gaussian beam (I dosage:570 J/cm2 @x⫽0). 共a兲 linear plot; 共b兲 log–log plot. Closed circles and solid lines indicate the experimental data and corresponding theoretical curves, respectively.

Bian et al. reported that a central peak appears in the PSR profile brought by the irradiation of a 2D Gaussian beam when the light intensity is very high 共⬎tens of W/cm2兲.9 Also the PSR gratings whose phase is shifted by ␲ from the ordinary gratings are observed in their experiment utilizing a sinusoidal beam of high intensity. They revealed that the azochromophores are optically bleached within the region irradiated with the light of high intensity, and the formed profile in such region becomes unerasable. For these ␲-phase shifts, Eq. 共8兲 deduced with respect to the ␤ value of the system can give a qualitative explanation reasonably; the accumulation of the light energy bleaches the azopolymer, and then the resultant decrease of ␣ turns negative ␤ into positive. If this scheme is valid, the ␲-phase shift found by Bian et al. should suggest the possibility of PSR formation on a transparent film although there has never been a report on PSR formation using solid transparent materials. Sanchez et al. succeeded to form PSR on liquid crystal azopolymer film even by the interfering beams of He–Ne laser (␭ ⫽633 nm) under the irradiation by incoherent bias light which can isomerize azochromophores.21 Most recently, as a result of the experiment using a pulse laser, Baldus et al. reported that the PSR can be formed even on amorphous polymers doped with chromophores which do not undergo any photoisomerization.22 Although the mechanism of the PSR formation caused by the pulse laser might be different from that by the cw laser, these experimental results suggest that the light providing the PSR driving force does not necessarily induce trans-cis-trans isomerization of azochro-

Sumaru et al.

mophores and supports that we treat the generation of the driving force and plasticization of an azopolymer independently in our model. As a result of experiments carried out by us and by Barrett et al.,10,18,19 the rate of PSR formation was found to have h 3 dependence on condition of small h. Our model, in which the photoinduced driving force is supposed to act as a body force on a viscous medium, explains this point reasonably. This is very important in proving that the hydrodynamic interaction is dominant in the mass transfer involved in PSR formation. When the mass transfer is brought by the migration of the chromophores in still matrix, the system will be described well by the simple incompressibility model by Bian et al. and h 1 dependence will be estimated for the rate of PSR formation.9 Possibly the similar tendency might be observed in the experiment using dye-doped polymer films. Further, supposing that only the film surface is participated in the mass transfer, the rate of the PSR formation will be independent of h and this estimation is inconsistent with most experimental results obtained for amorphous azopolymer films. In Sec. II, we supposed that the viscosity is uniform and independent of light intensity.9,10 However, amorphous azopolymers which are often used in the experiment on PSR formation are solid at room temperature. Concerning this fact, it can safely be said that the irradiation of light does plasticize the azopolymer quite effectively as many researchers have suggested.6,8,10,12,19 On the condition that the viscosity of the azopolymer is much dependent on light intensity I, it will be expected that the PSR profile is changed much according to I. More concretely, due to the restricted mass transfer in the dark region among the peaks of I, small depressions are expected to be found where peaks are supposed to exist usually in the PSR profile on the condition that a bias light is reduced sufficiently. However, this effect is not apparent since the threshold intensity seems to be relatively low and it is difficult to remove a bias light completely in the practical experiment. Although the connection to this effect of I-dependent viscosity is left unclear, Baldus et al. observed a unique PSR profile accompanied by the depressions as described earlier in the experiment of sinusoidal PSR formation on an amorphous azopolymer film using pulse laser.22 To this PSR profile, they tried to give an explanation in relation to photoinduced permittivity changes. On the other hand, this intensity dependence of viscosity should bring an inevitable effect on the viscosity distribution along depth direction regardless of the bias light intensity; on the condition that the absorption coefficient of an azopolymer is 10 ␮m⫺1, the light intensity is substantially zero in the region deeper than 1 ␮m from the surface. Therefore, the azopolymer in this deep and dark region is expected to behave as a fixed boundary binding azopolymer in a shallower region in practice. In our model assuming a uniform viscosity, however, even the azopolymer in the deep region is treated as a fluid buffering the binding effect of the substrate. Based on this assumption, it is suggested that the h-dependent PSR profile brought by a 1D Gaussian irradiation with a Gaussian width of 2 ␮m is obviously changed when h becomes larger than 5 ␮m as shown in Fig. 6. There-


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fore, it should be difficult to confirm this change in the PSR profile corresponding to the variation of h in the experiment. For observing the obvious change in the PSR profile in the range of h⬍1 ␮ m, our model requires a Gaussian width of 0.5 ␮m. Another important feature of the PSR phenomena is the fact that formed relief is erasable by anealing or by the irradiation of uniform light. This suggests that surface tension becomes dominant when the system is plasticized and the PSR driving force is absent. Although we ignored this surface tension in the model assuming that the surface modulation is small in the early stage of SRG formation, this treatment will not be acceptable when the modulation is large enough. On the other hand, Jiang et al. reported quite an interesting fact that even the uniform irradiation of light can increase the PSR grating amplitude instead of reducing it under a specific combination of polarization conditions in sinusoidally distributed light forming PSR gratings and uniformly distributed light irradiating it thereafter.13 This suggests that PSR gratings memorize the polarization condition of light in the PSR formation and are much influenced by it under the irradiation of uniformly distributed light thereafter. Although it is difficult to give a full explanation to this experimental result at this stage, the existence of some mechanism to produce the sinusoidal intensity distribution in the deeper region of the film is suggested. With respect to this point, locally varied absorbance due to the sinusoidal distribution in the orientation of azochromophores at the surface and in the film thickness may play an important role.

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ACKNOWLEDGMENTS

This work was supported by Industrial Technology Research Grant Program in ’00 from the New Energy and Industrial Technology Development Organization 共NEDO兲 of Japan, to which our sincere gratitude is due. APPENDIX: COMPLEX ELECTRIC SUSCEPTIBILITY OF ABSORBING MEDIUM

When the complex electric susceptibility of ith phase composing azopolymer ␹ i is expressed with its real part ␹ i⬘ and the imaginary part i ␹ ⬙i as follows:

␹ i ⫽ ␹ i⬘ ⫺i ␹ i⬙ ,

共A1兲

the optical electric field E progressing along the y direction satisfies the following one-dimensional Maxwell equation:

⳵ 2 E ␹ i⬘ ⫺i ␹ i⬙ ⫹1 ⳵ 2 E ⫺ ⫽0. ⳵y2 c2 ⳵t2

共A2兲

On the other hand, the optical electric field of the incident light in ith phase is expressed generally as

冋冉

E⫽E0 exp icK t⫺

niy c

冊册冉冊 exp ⫺

␣iy . 2

共A3兲

where E0 , K, n i , and ␣ i are the amplitude and the wave number of optical electric field in vacuo, the refractive index and absorption coefficient in ith phase, respectively. Substituting this equation into Eq. 共A2兲, the following relation among the complex susceptibility, ␣ i and K is obtained:

冉

␣i ⫹in i K 2

冊

2

⫹K 2 共 ␹ i⬘ ⫺i ␹ i⬙ ⫹1 兲 ⫽0.

共A4兲

Arranging this equation by the real parts and the imaginary parts, Eq. 共5兲 is derived finally.

V. CONCLUSIONS

P. Rochon, E. Batalla, and A. Natansohn, Appl. Phys. Lett. 66, 136 共1995兲. D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, Appl. Phys. Lett. 66, 1166 共1995兲. 3 A. Natansohn and P. Rochon, Adv. Mater. 11, 1387 共1999兲. 4 X. L. Jiang, L. Li, J. Kumar, D. Y. Kim, V. Shivshankar, and S. K. Tripathy, Appl. Phys. Lett. 68, 2618 共1996兲. 5 N. K. Viswanathan, S. B. Balasubramanian, L. Li, S. K. Tripathy, and J. Kumar, Jpn. J. Appl. Phys., Part 1 38, 5927 共1999兲. 6 J. Kumar, L. Li, X. L. Jiang, D.-Y. Kim, T. S. Lee, and S. Tripathy, Appl. Phys. Lett. 72, 2096 共1998兲. 7 S. Bian, L. Li, J. Kumar, D. Y. Kim, J. Williams, and S. K. Tripathy, Appl. Phys. Lett. 73, 1817 共1998兲. 8 N. K. Viswanathan, S. Balasubramanian, L. Li, J. Kumar, and S. K. Tripathy, J. Phys. Chem. B 102, 6064 共1998兲. 9 S. Bian, J. M. Williams, D. Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy, J. Appl. Phys. 86, 4498 共1999兲. 10 C. J. Barrett, P. L. Rochon, and A. L. Natansohn, J. Chem. Phys. 109, 1505 共1998兲. 11 T. Fukuda, H. Matsuda, N. K. Viswanathan, S. Tripathy, J. Kumar, T. Shiraga, M. Kato, and H. Nakanishi, Synth. Met. 102, 1435 共1999兲. 12 T. Fukuda, K. Sumaru, T. Yamanaka, and H. Matsuda, Mol. Cryst. Liq. Cryst. 345, 263 共2000兲. 13 X. L. Jiang, L. Li, J. Kumar, D. Y. Kim, and S. K. Tripathy, Appl. Phys. Lett. 72, 2502 共1998兲. 14 C. J. Barrett, A. L. Natansohn, and P. L. Rochon, J. Phys. Chem. 100, 8836 共1996兲. 15 T. G. Pedersen, P. M. Johansen, N. C. R. Holme, P. S. Ramanujam, and S. Hvilsted, Phys. Rev. Lett. 80, 89 共1998兲. 16 P. Lefin, C. Fiorini, and J.-M. Nunzi, Opt. Mater. 9, 323 共1998兲. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

In this study, we established a theoretical model for PSR formation covering thoroughly from the origin of the driving force to the dynamic process forming the surface relief by investigating the complex electric susceptibility and the hydrodynamic properties of azopolymer in detail. Experimental data we currently obtained and many experimental results reported in former literature were reproduced well by this model strongly supporting several hypotheses set up in this study; 共1兲 the PSR driving force is generated by the interaction between absorbing bodies and optical electric field, 共2兲 the azopolymer is plasticized by light irradiation and behaves as a viscous fluid, 共3兲 the PSR driving force acts on the azopolymer as body force, and 共4兲 the PSR is formed as a result of bulk deformation of the azopolymer. Indeed some of the experimental results reported by many research groups are still left beyond our theoretical framework and suggest that the processes which are not taken into account here may play a role in some situations. However, we believe that the theoretical model established here is one of the most feasible at this stage and that this should be an important theoretical foundation in revealing the whole picture of PSR formation in the future.

1 2

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P. W. Smith, A. Ashkin, and W. J. Tomlinson, Opt. Lett. 6, 284 共1981兲. K. Sumaru, T. Yamanaka, T. Fukuda, and H. Matsuda, Appl. Phys. Lett. 75, 1878 共1999兲. 19 T. Fukuda, H. Matsuda, T. Shiraga, T. Kimura, M. Kato, N. K. Viswanathan, J. Kumar, and S. K. Tripathy, Macromolecules 33, 4220 共2000兲. 17 18

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J. Appl. Phys., Vol. 91, No. 5, 1 March 2002 20

D. Y. Kim, L. Li, X. L. Jiang, V. Shivshankar, J. Kumar, and S. K. Tripathy, Macromolecules 28, 8835 共1995兲. 21 C. Sanchez, R. Alcala, S. Hvilsted, and P. S. Ramanujam, Appl. Phys. Lett. 77, 1440 共2000兲. 22 Baldus, A. Leopold, R. Hagen, T. Bieringer, and S. J. Zilker, J. Chem. Phys. 114, 1344 共2001兲.
