On the Inscription of Period and Half-Period ... - ACS Publications

4526

J. Phys. Chem. B 2008, 112, 4526-4535

On the Inscription of Period and Half-Period Surface Relief Gratings in Azobenzene-Functionalized Polymers Anna Sobolewska and Andrzej Miniewicz* Institute of Physical and Theoretical Chemistry, Wroclaw UniVersity of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland ReceiVed: January 3, 2008; In Final Form: February 14, 2008

Laser-light-induced surface relief grating inscription was carried out in the newly synthesized azobenzenefunctionalized poly(amide-imide)s having the same main- and side-chain structures but different substituents in the azobenzene groups. The gratings were inscribed employing the two-wave mixing technique with linearly polarized laser beams. Three different polarization configurations were used: s-s, p-p, and s-p. The relatively deep surface relief gratings of period Λ were formed for the case of s-s and p-p polarizations, whereas the s-p inscription resulted in the half-period grating (Λ/2) with the weak surface modulation. The origin of the formation of Λ/2 structure for s-p configuration results from the interference between zeroth- and first-order beams scattered on the polarization refractive index grating and having the same polarization. On the basis of this idea, we presented the simple kinetic model predicting and modeling the half-period grating formation with its temporal evolution. The proposed model is consistent with the experimental findings.

1. Introduction Surface relief gratings (SRGs), holographically inscribed on amorphousazopolymerfilms,havebeenextensivelyinvestigated1-18 since the first reports announced independently by two research groups of Rochon and co-workers19 and Tripathy et al.20 in 1995. They were fabricated in azopolymers using two interfering coherent laser beams, without any pre- and/or postprocessing procedure, and were inscribed at room temperature, i.e., well below the glass transition temperature (Tg) of these polymers.4,5 The basic phenomenon underlying the SRGs formation in azobenzene-functionalized polymers is the ability of azobenzene chromophores to undergo many successive reversible transcis-trans photoisomerization cycles which for linearly polarized light results in the permanent molecular reorientation of trans molecules perpendicular to the polarization plane. The chromophore reorientation is linked with a macroscopic polymer chain migration which is observed as a free surface modulation. The large modulation depth surface relief gratings were successfully inscribed on azo-functionalized polymer thin films.2,6,18 Because of the hundreds of nanometers deep surface modulation, the very high light diffraction efficiency could be observed in gratings inscribed in azopolymers. During holographic recording, prior to the surface relief grating formation, the amplitude and the phase gratings arise in azopolymers because of the volumetric absorption coefficient and refractive index modulations, respectively.3,6,13,21-23 The light diffraction efficiencies on these last two gratings are usually small when compared to the diffraction efficiency related to the surface relief grating. A lot of work devoted to the SRGs characterization has been done.2,24-28 Scientists considered possible mechanisms of the surface relief gratings formation and pointed out their potential applications. Main models explaining the origin of the surface relief gratings formation under spatially sinusoidal illumination are listed below: * Corresponding author: phone (048) 71-320-35-00, fax (048) 71-32033-64, e-mail [email protected].

(1) The free volume model proposed by Barrett and coworkers2,24 in which the driving force responsible for the mass transport was assumed to arise from pressure gradients induced by photoisomerization of azobenzene groups. Resulting viscoelastic flow of the material from the high-pressure to the lowpressure areas leads to the sinusoidal SRGs and was modeled using the Navier-Stokes equation.24 (2) The field gradient force model proposed by Tripathy et al.6,25 is based on the forces originating from the optically induced electric field gradient. The movement of polymer chains depends on the spatial variation of the material susceptibility (due to light-induced birefringence and dichroism), the optical field, and the field gradient along the grating wave vector K. This model nicely explains the light polarization dependence of the surface relief formation.6 (3) The asymmetric diffusion model proposed by Lefin and co-workers26,27 relates SRG formation with the creation of concentration gradients. The essential feature of the model is that dye-molecules undergo a 1-D random walk along their photoexcitation direction, thus inducing a net flux of molecules out of illuminated regions toward the darker regions. (4) The mean-field theory proposed by Pedersen28 assumes that chromophores are subjected to anisotropic intermolecular interactions. The mean field of oriented by light chromophores tend to align other chromophores in the same direction and cause an attractive force between side-by-side chromophores oriented in the same direction causing their order and aggregation. This mechanism properly describes the surface relief formation in side-chain azobenzene liquid crystalline polymers, which show an in-phase SRGs; it means that surface profile maxima are coincident with the maxima of the light intensity pattern. It should be pointed out that such behavior is in contrast to amorphous polymers, where the reverse effect is observed; i.e., the light intensity maxima correspond to surface minima. In this paper we report on a surface relief gratings inscription for three different types of linear polarization configurations: s-s, p-p, and s-p using a two-wave mixing setup. The

10.1021/jp800048a CCC: $40.75 © 2008 American Chemical Society Published on Web 03/27/2008

Relief Gratings in Azobenzene-Functionalized Polymers

J. Phys. Chem. B, Vol. 112, No. 15, 2008 4527

Figure 1. Chemical structure of a series of poly(amide-imide)s and their glass transition temperatures Tg (DSC method). Various terminal groups R influence the polarity of the chromophore.

investigations were performed in a series of newly synthesized azobenzene-functionalized polymers poly(amide-imide)s (PAIs). Similar studies, namely, the influence of the configuration of the recording beams polarization on the SRGs formation in azopolymers, have already been performed. Tripathy and coworkers had made detailed analysis of how different polarization geometries influence the amplitude of the surface modulation.6 Lagugne-Labarthet et al.14,22 investigated the light diffraction efficiency, birefringence, and the surface modulation induced by linearly polarized beams, a combination of (45° and circularly polarized beams, and co- and contracircularly polarized beams. Results reported here will be compared and discussed in relation to these reports. In particular, we focused our attention on unusual effect of formation of peculiar half-period SRG gratings when s-p polarization geometry was used. This phenomenon has already been reported by Lagugne-Labarthet and co-workers29 and explained as a result of the additional interference between zeroth- and first-order beams having the same output polarization in s-p geometry. In order to describe this effect, they applied a phenomenological approach based on Jones’ matrix analysis. Following their idea, we present the simple kinetic model for the half-period grating formation in s-p configuration. Numerical simulations presented here qualitatively confirm the half-period grating formation evolution and agree with SRG shapes measured by atomic force microscopy (AFM) for that configuration. 2. Experimental Section 2.1. Materials. The SRGs inscription was performed on the thin films of poly(amide-imide)s containing azobenzene moieties in the side-chain positions. The films were prepared by casting a solution of PAIs in a polar solvent N-methyl-2pyrrolidone onto the cleaned glass substrates. The polymer film thicknesses (d0) were determined with the help of an interference microscopic method of Tolansky. Poly(amide-imide)s have been synthesized from diamide dianhydrides and diamines with various substituents in the azobenzene group using the high-temperature polycondensation method. The chemical structure of poly(amide-imide) is shown in Figure 1, where R denotes a substituent in the azobenzene unit. Details about the synthesis and the polymer characterization can be found elsewhere.30 2.2. Optical Setup. Surface relief gratings were recorded using a standard degenerate two-wave mixing technique (DTWM). The basic experimental setup of DTWM is shown in Figure 2. The linearly polarized beam coming from the laser source is split into two beams of the same polarization. In the simplest case both beams are s-polarized (s-s polarization geometry). L,s When such polarized beams, with intensities IR,s inc and Iinc, cross

Figure 2. Degenerate two-wave mixing experimental setup: L ) laser, BS ) beam splitter, M ) mirror, PF ) polymer film, D ) detector, PC ) personal computer.

each other in the polymer film, a simple interference fringe pattern along the x-direction is formed:31

I(x) ) I0(1 + m cos(Kx))

(1)

L,s where I0 ) IR,s inc + Iinc, m is the modulation factor m ) 2

L,s R,s L,s xIR,s inc Iinc/(Iinc + Iinc), and K ) 2π/Λ is the interference pattern

wave vector. If the optical properties of a polymer film are changed accordingly to this pattern, a diffraction grating appears, and then Λ is the period of the light-induced diffraction grating in the material Λ ) λ/2 sin(θ/2), where λ is the wavelength of the writing beams and θ is the intersection angle between the beams. The power of light diffracting on the grating into the first-order diffraction direction serves as a measure of a temporal evolution of the grating buildup process. The reported here SRGs were inscribed for the three qualitatively different polarization configurations s-s, p-p, and s-p, where s and p stand for light electric field perpendicular and parallel to the incidence plane, respectively. The direction of the electric field of the optical waves is perpendicular to the x-direction for s-s polarization geometry. There is a light intensity modulation along the x-axis in both cases of p-p and s-s geometries as described by eq 1. However, in the s-p configuration the light intensity is constant along the x-direction, L,p I(x) ) IR,s inc + Iinc , but periodic changes of generally elliptical polarization occur along the grating wave vector K. All polarization geometries together with the incident light polarization modulations along the grating period Λ are shown in Table 1. The full Jones matrix description of these configurations is given in ref 22. All SRGs analyzed in this work were recorded under the same experimental conditions. The light beam of wavelength λ ) 514.5 nm from a CW Ar+ laser (Innova 90, Coherent) was used for grating inscription. The light intensities of the two laser beams measured at sample surface were kept equal IRinc ) ILinc ) 570 mW/cm2, and 1 h exposure time was used. The angle between the writing beams was fixed at θ ) 10°, resulting in a

4528 J. Phys. Chem. B, Vol. 112, No. 15, 2008

Sobolewska and Miniewicz

TABLE 1: Periodic Polarization Modulation of the Light Used for the Grating Inscription for Different Polarization Geometries: s-s, p-p, and s-pa

a The polarization states are given according to the reference axes (x, y) and are reported as a function of the phase difference δ between the two interfering pump beams. Here I is a sum of intensities of the interfering beams.22

TABLE 2: Surface Relief Modulation Amplitudes ∆d and Maximum Diffraction Efficiencies η1 in Poly(amide-imide)s Measured for s-s, p-p, and s-p Polarization Configurations polarization configurations s-s polymer

d0 [nm]

∆d [nm] Λ

1a 1b 1c 1d

2140 1290 1040 1620

34 30 44 48

p-p η1 [%]

∆d [nm] Λ

0.58 0.46 0.99 0.62

85 62 69 73

s-p η1 [%]

∆d [nm] Λ

∆d [nm] Λ/2

η1 [%] Λ

1.66 1.06 0.96 0.93

33 18 12 13

15 8 7 9

1.30 0.42 0.23 0.76

spatial periodicity of Λ = 3 µm. The polymer film was placed perpendicularly to the bisector of the two incident beams in order to ensure that the grating wave vector K is lying within the polymer film. The temporal evolution of the first-order diffracted beam’s power was measured using two calibrated silicon detectors (LM-2 Silicon HD Smart Sensor, Coherent) connected to a PC via the power meter Laser Pad system. The first-order diffracted light power temporal evolution was then used for determination of the first-order diffraction efficiency in function of time (η(1(t) ) I(1(t)/Iinc). 3. Results and Discussion Surface relief gratings inscribed on the polymer films were observed with the help of atomic force microscopy in tapping mode. Results of the surface modulation measurements for all investigated poly(amide-imide)s, obtained for the abovementioned three polarization geometries, are gathered in Table 2. The values of the maximum first-order diffraction efficiencies reached within the inscription time and the polymer film thicknesses are also reported. The surface reliefs were analyzed for all studied polymers. Exemplary surface profiles of SRGs inscribed in polymer 1a under the three studied polarization configurations are shown in Figure 3. The largest surface amplitudes in PAIs were recorded under the p-p polarization configuration irrespectively of different substituents in the azobenzene group (cf. Table 2). Smaller amplitudes were observed for the s-s polarization geometry and the smallest for the s-p one. Interesting results were obtained for the s-p geometry, where besides the surface relief grating with period Λ a grating with a half-period appeared. Smaller spatial surface modulation of Λ/2 periodicity is clearly observed by AFM technique (cf. Figure 3c), and it was observed for all studied poly(amide-imide)s. The halfperiod grating formation is characteristic for the s-p polarization

Figure 3. Atomic force microscopy scans (13.1 × 13.1 µm2) of the surface of poly(amide-imide) 1a for the three polarization geometries: s-s (a), p-p (b), and s-p (c). 2D and 3D views to the left and right, respectively. The surface profiles of these gratings are shown in the middle. Lines on pictures indicate places where the profiles were measured.

configuration as was already noticed and analyzed in refs 29 and 32. In order to qualitatively and quantitatively describe this effect, the simple kinetic model will be proposed in the next section. The results of numerical simulations of half-period grating formation will also be shown.

Relief Gratings in Azobenzene-Functionalized Polymers The discussion of the results obtained by us will base on their comparison with the earlier reports6,22,25 agreeing that, among s-s, s-p, and p-p polarization geometries, the largest SRG’s amplitude, thereby the highest diffraction efficiency, is obtained for the p-p geometry. Our results obtained in PAI’s confirmed these findings. The significantly smaller surface modulations and lower diffraction efficiencies were measured for the s-s and s-p configurations than for the p-p one. However, Tripathy et al.6 reported that the smallest surface modulation is observed when the two s-polarized beams are used (∆d < 10 nm) while s-p gives higher amplitudes (∆d < 20 nm). This is consistent with the field gradient force model6,25 which predicts that no force is acting on polymer chains in the s-s configuration, and hence no appreciable SRG’s formation is predicted. The results presented by Lagugne-Labarthet et al.22 are qualitatively different because the weakest surface amplitude (∆d ) 2.5 nm) was observed for two orthogonally polarized beams s-p. The reason cannot be explained solely on the basis of different intensities used for grating inscription: 50 mW/cm2 in Tripathy and co-workers6 and 90 mW/cm2 in Lagugne-Labarthet et al.22 Results obtained by us in poly(amide-imide)s seem to agree with that reported by Lagugne-Labarthet group; i.e., more efficient surface modulation occurs for s-s geometry than for s-p one (cf. Table 2). We believed that the reason underlying the observation of large surface modulation amplitude in the s-s configuration was a relatively strong irradiance level. The maximum light intensity modulation ∆I in our experiment was R,s 2 ∆I ) 2.3 W/cm2 (IR,s inc ) 570 mW/cm and ∆I ) 4Iinc ). Therefore, the thermal effects facilitating the polymer mass transport are highly probable and can be responsible for considerable surface modulation in this geometry. The weakest surface relief amplitude modulation occurred for s-p configuration where the half-period gratings were observed for all PAIs. The appearance of such a structure was mentioned by Tripathy et al. in ref 6, and the detailed studies of the half-period grating formation were reported by LagugneLabarthet et al.29 It is worth noting that despite the smallest surface amplitude obtained in s-p recording geometry, the relatively high diffraction efficiencies were measured when compared to those observed for s-s configuration (cf. Table 2). The explanation had already been pointed out by LagugneLabarthet et al.,22 who concluded that in the case of s-p polarization geometry the light-induced birefringence, i.e., modulation of refractive index in the volume of the polymer ∆n, is maximum when compared to other used geometries. The dynamics of grating recording depended on the polarization geometry. The exemplary experimental curves of the firstorder diffraction efficiency in function of time, measured during the grating inscription for PAI 1a for s-s, p-p, and s-p polarization combinations are shown in Figure 4. When the two s-polarized pump beams were turned on, the dynamics of the diffraction efficiency increased up to the saturation level. Saturation was observed after about 2500 s, giving the maximum diffraction efficiency η1 ) 0.58%. In the case of p-p polarization geometry a very slow process of grating buildup was observed. The curve of grating recording displayed a nearly linear variation up to η1 ) 1.66% after 3500 s. Up to this time of inscription the plateau was not observed; thus, for prolonged exposure the larger surface relief modulation could possibly be obtained. For orthogonal beam polarization (s-p) the recording curve showed rapid increase at a relatively short time and then after about 300 s of irradiance the saturation was reached. It remained at the same level to the end of recording, i.e., 3500 s, showing diffraction efficiency around


Figure 4. Dynamics of the grating recording in the polymer PAI 1a observed via self-diffraction process. Curves η1(t) recorded for s-s (a), p-p (b), and s-p (c) polarization geometries.

η1 ) 1.3%. Temporal evolutions of the grating inscription for other studied poly(amide-imide)s were qualitatively similar. Concluding this part of the paper, we state that the most efficient surface modulation was observed for the p-p geometry which is in agreement with other reports. Clear conclusion about which of the other studied geometries, i.e., s-s and s-p, leads to more efficient surface modulation cannot be decided. Finally, presented results confirmed that the half-period grating appears when the s-p polarization combination is used. At the end of this section it should be stressed that we are aware of the fact that the comparison of the experimental results presented here with those reported by other groups is not straightforward. It is mainly due to the differences in investigated materials and the way of their preparation.



4. Half-Period Grating Formation When azobenzene-functionalized polymer film is irradiated by the interference pattern I(x) of the writing beams, sinusoidally modulated along the x-axis (s-s or p-p) (cf. eq 1) or alternatively by a light polarization pattern (s-p) (cf. Table 1), the spatial modulations of the refractive index n(x,t) and the thickness d(x,t) occur in the material:33

∆n(t) cos(Kx) 2

(2)

∆d(t) cos(Kx + ∆φ(t)) 2

(3)

n(x,t) ) n0 + d(x,t) ) d0 +

where ∆n(t)/2 and ∆d(t)/2 are the amplitudes of the bulk refractive index and the surface relief gratings, respectively, n0 is the average refractive index of the polymer at the respective light wavelength λ, and ∆φ (t) is the possible phase shift between the gratings. During DTWM experiment behind the polymer sample the multiple-order light self-diffraction is observed (i.e., a Raman-Nath light scattering regime is fulfilled). In such case, for the pure phase grating the first-order diffraction efficiency η(1(t), defined as the ratio between the L(R) first-order diffracted beam intensity I(1 (t) and the input beam L(R) intensity Iinc , is described by the square of the first-order Bessel function J1:31

η(1(t) )

L(R) (t) I(1

IL(R) inc

) J12(∆φ(t))

(4)

where ∆φ(t) is the maximum phase retardation accumulated by a plane wave of wavelength λ transmitted trough the system. For the purpose of clarity of presentation we neglected here the light absorption effect and an absorption grating. The former influences only the magnitude of observed diffraction efficiency, and the latter requires different mathematical treatment. Neglecting of absorption grating in the case of s-p polarization configuration is justified by its relatively small amplitude (no bleaching is expected for uniform light intensity) and small contribution to the light diffraction. Assuming that there is no phase shift (∆φ(t) ) 0) between the gratings ∆φ(t) is a sum of the phase retardations ∆φn(t) and ∆φd(t) due to the bulk refractive index (∆n(t)) and the surface relief (∆d(t)) gratings, respectively:33

2π∆n(t)d0 ∆φn(t) ) θ λ cos 2

()

∆φd(t) )

2π∆d(t)neff θ λ cos 2

()

Figure 5. Scheme of polarization states of the two zeroth-order and two first-order light beams observed behind the sample of azo-polymer in the self-diffraction experiment during grating recording under s-p polarization geometry.

where τ1 and τ2 are the characteristic grating formation time constants of the above-mentioned gratings and ∆nmax and ∆dmax are the maximum refractive index and thickness modulations, respectively. Let us now consider the grating inscription using s-p configuration, for which, as we stated before, there is no light intensity modulation along the grating period I(x) ) const ) I0 L,p (I0 ) IR,s inc + Iinc ), i.e., modulation factor m ) 0 (cf. eq 1). Furthermore, during s-p polarization grating recording in azopolymers the light polarization of the beam diffracted into zeroth-order direction has the same linear polarization as the incident beam whereas the light polarizations of the beams diffracted into (1 orders have polarizations orthogonal to the incident one.34,35 This unique feature is illustrated in Figure 5. In function of the grating inscription time the multiple diffraction orders appear which can reach roughly 1% of the incoming light intensity. Since the s-p polarization geometry is considered as the one of the least effective in the SRG formation,22 the measured light diffraction comes from the modulation of the refractive index in the bulk of the polymer. Therefore, we can assume that the contribution of the phase retardation due to surface relief grating in the maximum phase retardation ∆φ(t) (cf. eq 4) is negligible. Thus, the first-order diffraction efficiency sp η(1 (t) can be expressed by (cf. Figure 5) sp η(1 (t)

(5)

)

L,p I+1 (t)

IL,s inc

)

R,s I-1 (t)

IR,p inc

) J12(∆φn(t))

(9)

)

(10)

and after taking into account eq 6:

(6)

where neff is the effective refractive index associated with a corrugated surface layer (polymer + air) and is estimated using the Maxwell-Garnett approach to amount neff ≈ (n0 + 1)/2.6,34 The dynamics of buildup of the refractive index and surface relief gratings, ∆n(t) and ∆d(t), respectively, can be described by the single-exponential growth functions:

∆n(t) ) ∆nmax[1 - exp(-t/τ1)]

(7)

∆d(t) ) ∆dmax[1 - exp(-t/τ2)]

(8)

(

sp η(1 (t) ) J12

2π∆nsp(t)d0 θ λ cos 2

()

In eq 10 the dynamics of ∆nsp(t) is described by the exponential growth function, shown in eq 7, with time constant τ1,sp and the maximum refractive index grating ∆nmax,sp. Taking into account only the strongest first orders described sp (t), for prolonged grating recordby diffraction efficiency η(1 ing times, there must also occur an interference between these orders that have the same polarizations s-s and p-p (cf. Figure R,s 5). Therefore, the beam IL,s 0 will interfere with the I-1(t) and R,p L,p I0 will interfere with the I+1 (t) one. Again we recall the fact that the surface relief grating formation strongly depends on


J. Phys. Chem. B, Vol. 112, No. 15, 2008 4531 and, after taking into account eqs 5 and 6, can be rewritten to the final forms:

( (

ss η-1 (t) ) J12

pp η+1 (t) ) J12

() ) () )

2π∆nss(t)d0 2πneff∆dss(t) + θ θ λ cos λ cos 2 2

()

2π∆nss(t)d0 2πneff∆dpp(t) + θ θ λ cos λ cos 2 2

()

(13)

(14)

In eqs 13 and 14 the dynamics of buildup of the refractive index grating (∆nss(t)) and the surface relief gratings (∆dss(t) and ∆dpp(t)) are described by the exponential growth functions (cf. eqs 7 and 8) with different grating formation time constants: τ1,ss for ∆nss(t), τ2,ss for ∆dss(t), and τ2,pp for ∆dpp(t). The value of the maximum refractive index grating ∆nmax is assumed to be the same in the both polarization geometries (∆nmax,ss ) ∆nmax,pp), whereas the maximum thickness modulations ∆dmax,ss and ∆dmax,pp for s-s and p-p configurations, respectively, are different. Knowing the diffraction efficiencies for the first orders in sp function of time η(1 (t), it is easy to calculate the modulation factors mss(t) and mpp(t) for the interference between the zerothand first-order diffracted beams having the same polarization during s-p recording. The interference occurs between IL,s inc and R,s L,p (t) for s-s geometry and between IR,p and I (t) for p-p I-1 inc +1 (cf. Figure 5); therefore, the mss(t) and mpp(t) can be written as follows:

mss(t) )

mpp(t) )

Figure 6. Plots of temporal evolution of the first-order diffraction ss pp sp efficiencies: η-1 (t) for s-s (a), η+1 (t) for p-p (b), and η(1 (t) for s-p (c) recording geometries, based on parameters given in the text.

the polarization of the recording beams. It was experimentally proved, in this study, that the p-p configuration is more efficient than the s-s one. Therefore, for the same inscripss pp (t) and η+1 (t) for s-s tion times the diffraction efficiencies η-1 ss pp and p-p geometries are different (η-1(t) < η+1(t)) according to the different surface relief phase retardations (∆φd,ss(t) and ∆φd,pp(t)) that they produce (∆dss(t) < ∆dpp(t)). Assuming that (i) the phase contribution due to the refractive index changes in the bulk ∆φn(t) is the same for s-s and p-p ∆φn,ss ) ∆φn,pp and (ii) there is no phase shift between the bulk refractive index and the surface relief gratings ∆φs(t) ) ss pp ∆φp(t) ) 0, the η-1 (t) and η+1 (t) can be written as (cf. Figure 5) ss η-1 (t)

)

pp (t) ) η+1

R,s (t) I-1

IR,p inc L,p I+1 (t)

IL,s inc

) J12(∆φn,ss(t) + ∆φd,ss(t))

(11)

) J12(∆φn,ss(t) + ∆φd,pp(t))

(12)

R,s 2xIL,s incI-1(t) R,s IL,s inc + I-1(t) L,p 2xIR,p inc I+1 (t) L,p IR,p inc + I+1 (t)

)

)

sp R,p 2xIL,s incη-1(t)Iinc sp R,p IL,s inc + η-1(t)Iinc sp L,s 2xIR,p inc η+1(t)Iinc sp L,s IR,p inc + η+1(t)Iinc

(15)

(16)

R,p For the equal light intensities of the incident beams IL,s inc ) Iinc we have mss(t) ) mpp(t). R,s The interference between the IL,s inc and I-1(t) and between the R,p L,p Iinc and I+1 (t) leads to respective surface modulations described by dss(x,t) and dpp(x,t) (cf. eq 3)

dss(x,t) ) d0 +

∆dss(t) mss(t) cos(2Kx - δs) 2

(17)

dpp(x,t) ) d0 +

∆dpp(t) mss(t) cos(2Kx - δp) 2

(18)

Notice that the grating wave vector is now doubled 2K (K ) L,p 2π/Λ). This is due to fact that the angle between IR,p inc and I+1 (t) R,p L,p (Iinc and I+1 (t)) equals 2θ (cf. Figure 5). Symbols δs and δp describe the phase shifts between the gratings of period Λ/2, namely, dss(x,t) and dpp(x,t), and the grating of period Λ. They may result from the fact that the Λ/2 grating wave vectors 2K are not parallel to the Λ grating wave vector K, and they depend on the angle θ. In Figure 3c one can notice the presence of both gratings but with different surface modulations: larger for Λ and smaller for Λ/2. We attribute the formation of the Λ SRG during s-p grating recording to the interference of the two incident beams s and p not perfectly orthogonally polarized. For the case of imperfect orthogonal beam polarization (which frequently occurs) some intensity modulation will take place



Figure 7. Calculated results of the surface deformation d(x,t) according to the presented model of Λ/2 grating formation for the s-p configuration. Graphs were obtained for symmetrical case for different times of grating recording: 50 (a), 200 (b), 500 (c), 1000 (d), 3000 (e) and 6000 s (f). Note that besides the basic period Λ ) 3 µm the structure with the period Λ/2 ) 1.5 µm is clearly visible.

with periodicity of Λ. In such a case the modulation factor msp(β) can be calculated according to

msp(β) )

2x1 cos(β) 1 + cos(β)

(19)

where β is the azimuthal angle between light polarization vectors s and p. According to eq 19 for perfect orthogonal beam polarization (β ) 90°) the modulation factor msp(β) in the interference pattern equals to 0, therefore there is no intensity modulation I(x) ) const. Assuming, that the beams s and p are not perfectly mutually orthogonal, for instance β ) 88°, the light intensity modulation with modulation factor msp (88°) ) 0.361 will appear. This light intensity modulation can lead to the surface modulation dsp(x,t) with period Λ:

dsp(x,t) ) d0 +

∆dsp(t) msp(β) cos(Kx) 2

(20)

where ∆dsp(t) expresses the dynamics of the surface relief grating formation with period Λ, and it is assumed to be described by the single-exponential function (cf. eq 8) with

growth time constant τ2,sp and with the maximum thickness modulation ∆dmax,sp. In summary, we can say that during grating recording using s-p configuration three possible interferences of optical filed can be distinguished. Two of them are the interferences of the incident beams with their first orders having the same polarization. They are responsible for the surface relief formation with Λ/2 period. The third one is a result of the intentionally or unintentionally introduced interference between the not orthogonally polarized s and p beams. This can lead to small but nonnegligible intensity modulation able to create SRG with a period of Λ. Summing up all three time- and polarizationdependent surface modulations, one can obtain the final surface modulation d(x,t) during the s-p grating recoding process in the form

d(x,t) ) dss(x,t) + dpp(x,t) + dsp(x,t) - 2d0

(21)

According to the model presented above the numerical simulations of the half-period grating formation were performed for the chosen set of parameters. Knowing the experimental results of the surface relief inscription in PAIs, we assumed the time



Figure 8. Results of the surface deformation d(x,t) calculated according to the presented model of Λ/2 grating formation when the symmetry is broken for 50 (a), 200 (b), 500 (c), 1000 (d), 3000 (e), and 6000 s (f) grating recording times.

constants τ1,sp ) 150 s, τ1,ss ) 200 s, τ2,ss ) 1000 s, τ2,pp ) 4000 s, τ2,sp ) 200 s and the maximum grating amplitudes ∆nmax,sp ) 0.018, ∆nmax,ss ) 0.015, ∆dmax,ss ) 0.035 µm, ∆dmax,pp ) 0.085 µm, ∆dmax,sp ) 0.030 µm. Other parameters were following: n0 ) 1.600, d0 ) 1 µm, neff ) 1.300, λ ) 514.5 nm, θ/2 ) 5°, δs ) -δp ) π/6. In Figure 6, we present ss pp sp (t), η+1 (t), and η(1 (t) the first-order diffraction efficiencies η-1 for s-s, p-p, and s-p geometries simulated according to eqs 13, 14, and 10. The dynamics of grating recording for these three configurations is comparable with our experimental findings (cf. Figure 4). However, there are differences in the values of diffraction efficiencies for s-s and p-p polarization geometries between the calculated ones and those measured experimentally. The reason for that lies in the assumption which ss pp (t) and η-1 (t) that there is was made for the simulation of η+1 no phase shift between the bulk refractive index and surface relief gratings (∆φs(t) ) ∆φp(t) ) 0). It was already proved that the possible phase shift can influence the diffraction efficiency and its dynamics.33 Using diffraction efficiency functions as input ones, we can perform simulation of surface relief grating amplitudes. From sp (t) the modulation factor mssfunctional dependence of the η(1 (t) was calculated (eq 15) and then the surface modulations dss(x,t) and dpp(x,t) (eqs 17 and 18). The surface modulation which

occurs for s-p configuration dsp(x,t) was also taken into account and simulated (eq 20). Finally, the total surface relief modulation d(x,t) was calculated according to eq 21. The numerical simulations were performed for two different cases (i) when the symmetry of the experiment is preserved, then the absolute values of phase shifts δs and δp are equal and (ii) when the symmetry is broken. The calculations were done (i) for δs ) π/6 ) -δp and (ii) for δs ) 2(π/6) and δp ) 0. In Figures 7 and 8 the surface relief modulations d(x,t) calculated for both (i) and (ii) cases for different recording times 50, 200, 500, 1000, 3000, and 6000 s are shown. The graphs in Figure 7 show the results for the case when the symmetry is preserved, and in Figure 8 they correspond to the case of broken symmetry. All calculations were performed with the help of the Mathcad program. For short recording times (

On the Inscription of Period and Half-Period ... - ACS Publications

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