6690
J. Phys. Chem. B 1999, 103, 6690-6699
Azopolymer Holographic Diffraction Gratings: Time Dependent Analyses of the Diffraction Efficiency, Birefringence, and Surface Modulation Induced by Two Linearly Polarized Interfering Beams F. Lagugne´ Labarthet,† T. Buffeteau, and C. Sourisseau* Laboratoire de Physico-Chimie Mole´ culaire, C.N.R.S., UMR 5803, UniVersite´ Bordeaux I, 351 cours de la Libe´ ration, 33405 Talence cedex, France ReceiVed: March 3, 1999
Dynamical experimental studies of the diffraction efficiency, birefringence, and surface relief modulation were carried out on functionalized azopolymer films of p(DR1M-co-MMA) with a 12% mole fraction of DR1M. The gratings were recorded using two linearly polarized pump beams (λ ) 514.5 nm) either with a parallel configuration (p+p) or (s+(-s)) or with orthogonal polarizations (p+s). A general Jones matricial approach in conjuction with real-time polarization analyses of the first order diffracted beam (S+1) appears to be a quite sensitive method to extract relevant parameters such as birefringence (∆n) and surface relief modulation (2∆d). The phase matrices have been developed considering the pump polarization of the two interfering beams and taking into account the orientational “angular hole-burning” model under low pump irradiances. Numerical calculations allow us to extract the time variations of ∆n and 2∆d and to compare the various efficiencies in grating formation. The obtained values for ∆n are then compared with polarimetric measurements and the basic mechanisms or driving forces in the relief formation are also discussed in connection with recently proposed phenomenological models.
Introduction Recent observations of surface relief modulation effects induced in functionalized azopolymers have opened a new class of holographic materials which are sensitive to the polarization state of the interfering writing pump beams.1-5 This is a consequence not only of the trans to cis isomerization with respect to the NdN double bond, which causes an angular reorientation of the rodlike trans azobenzene units in the glassy polymer matrix, but also of the existence of a modulated electromagnetic field impinning the sample, in particular along the grating vector. Such materials are very promising because of their high diffraction efficiency (up to 30-35% on the firstorder diffracted beam), and applications in slab wave guides have already been reported.6,7 In addition, when compared with standard holographic emulsions and photopolymers, a surface relief grating can be easily recorded without any developing processing, photocurring, or wet bleaching. Moreover, the surface relief formation occurs without any polymer ablation and the initial flat film can be recovered after heating up to the polymer Tg. Nevertheless, even though it is well-known that the photoinduced anisotropy induces a modulation of the refractive index, which causes a phase grating with a diffraction efficiency limited to a few percent, the surface modulation formation is more puzzling and several models have already been proposed in order to explain the basic mechanisms or driving forces responsible for its formation: (i) Rochon and Natansohn7-9 have proposed that the surface relief grating is mainly due to the volume change during the trans to cis isomerization process, the cis isomer requiring a larger volume * Author to whom correspondence should be addressed. E-mail: csouri@ morgane.lsmc.u-bordeaux.fr. † Present address: Royal Military College, Physics department, P.O. Box 17000, Stn Forces, Kingston, Ont K7K 7B4, Canada.
than the trans isomer. Considering the mechanism of an isomerization-driven free volume expansion to produce internal pressure gradients upon the amorphous polymer yield point, a model based on the Navier-Stockes equations has been proposed to describe the short time dynamics of the viscoelastic flow of the polymer resulting from the laser-induced isomerization in the bulky chromophores.9 (ii) The Tripathy and co-workers’ model10-12 is based on a gradient force model induced by the modulated polarized electromagnetic field; this approach explains nicely the various polarization effects due to a single laser pump beam and observed on such a polymer surface. (iii) Lefin et al.13,14 have proposed that the surface modulation could in part be due to a translational wormlike diffusion of the azobenzene chromophores from regions of higher isomerization to those of a lower rate, creating a concentration gradient responsible for diffraction. Such assumptions have been recently checked by micro-Raman measurements and polarized white light transmision confocal microscopies in our group.15 (iv) Todorov and Hvilsted16-19 have recently proposed a Jones matricial analysis in order to extract the contribution of the phase grating (linear and circular anisotropies) and of the topographic modulation induced in a side-chain azobenzene liquid crystalline polymer. In this respect, Pedersen and co-workers20 using a mean-field theory approach have demonstrated that chromophores are subject to anisotropic intermolecular interactions leading to a mass transport effect which depends strongly on the polarization of the two interfering beams. Independent of these pioneering works, we have recently proposed that a Jones’ matrix approach could be extensively developed to explain the dynamics of grating formation in amorphous polymer thin films.21 In a previous work, we have thus investigated the formation of birefringence and surface
10.1021/jp990752j CCC: $18.00 © 1999 American Chemical Society Published on Web 07/28/1999
Azopolymer Holographic Diffraction Gratings relief grating on both doped and functionalized polymer films irradiated using co- and contra-circularly polarized pump beams. In the different areas of the grating, we have considered a pure reorientation of the chromophores with respect to the resulting incident electric field directions. In fact, it appeared that the pure orientation model was not entirely verified, and from polarized micro-Raman measurements of the order parameters 〈P2〉 and 〈P4〉, we have demonstrated that the orientation of chromophores in the different regions was markedly modified by the surface relief formation.22 Even though the pure orientational effects are not sufficient to explain all the experimental results, it is clear that the Jones’matricial approach is a quite sensitive method to extract some relevant parameters and their time variations from polarization analyses of the diffracted orders. In the present work, we have thus extensively developed the same approach for gratings inscribed on a functionalized copolymer film using linearly (p+p) and (s+(-s)) and orthogonally (p+s) polarized pump beams, where p and s stand for laser electric field parallel and perpendicular to the incidence plane, respectively. The paper is thus organized as follows: After the Experimental Section, describing the optical setup used for grating inscription and for the time analyses of the diffraction efficiency, we present in a theoretical part the analytical expressions of the polarized intensities expected on +1 or -1 diffracted orders for different pump polarization configurations, namely, (p+p), (s+(-s)), and (p+s), respectively. In the Experimental Results, the time variations of the first-order diffracted beam during irradiation and relaxation periods are reported in the same order. Then, numerical solving strategies are discussed and extracted variations of the birefringence (∆n), of the surface relief (2∆d), and of their dephasing (φ0) are presented in each case. Finally, in the last section, the obtained results are compared with birefringence measurements and AFM (atomic force microscopy) experiments and some conclusions about the most efficient diffraction processes and the surface modulation formation mechanisms are discussed. Experimental Section We have used a functionalized copolymer system p(DR1Mco-MMA) with a 12% mol fraction of DR1M. Thin films were made by spin-casting a solution of copolymer in chloroform. Films are then baked for 1 h at 90 °C. All the starting films (0.9-1.1 µm thickness) have been checked by polarized UVvis measurements to be isotropic.23 The experimental setup for time dependent analysis of the diffraction efficiency is presented in Figure 1, and further details can be found elsewhere.21 Two optical setup were used for grating inscription. The first one, used for (p+p) and (s+(-s)) polarization configurations, is similar to that proposed by Rochon et al.7-9 and is shown in Scheme 1a. A 514.5 nm pump beam from a linearly polarized laser (ILT 5490) was used for writing. The beam was expanded to a diameter of about 10 mm and the irradiance on the sample was approximately 90 mW/ cm2. Using a half-wave plate, the polarization state was either in the plane of incidence (p) or out of the incidence plane (s). In the second setup, used for the (p+s) configuration (Scheme 1b), a Wollaston prism was included to separate with an angular aperture of about 20° two orthogonally polarized beams from the initial circularly polarized pump. The two orthogonal beams were then set in a parallel direction using a 100 mm focal lens. In both experiments the sample was set at a right angle to a front surface mirror. The mirror reflected half of the incident
J. Phys. Chem. B, Vol. 103, No. 32, 1999 6691
Figure 1. Experimental setup for grating inscription and time dependent analyses of the diffraction efficiency. For the probe beam, we have used the following configurations: rotator V + polarizer V (IVV), rotator V + polarizer H (IVH), λ/4 + polarizer V (I⊥), λ/4 + polarizer H (I|) and λ/4 without polarizer (Itotal). The symbols 0, (1 indicate the 0th transmitted and the (1 order diffracted beams, respectively.
SCHEME 1
a
The laser beam is linearly polarized in the incidence plane (p) or perpendicularly to the incidence plane (s) using a half-wave plate. The beam is then expanded to a diameter of 1 cm. b The pump beam is first circularly polarized and split into two beams with orthogonal polarizations and an angular aperture of 20° by using a Wollaston’s prism.
pump light which was interfering with the direct beam onto the sample. The sample holder was set on a goniometer in order to control the incident angle which defines the pattern spacing Λ according to Bragg’s law, Λ ) λ/2sin θ. In all the experiments, the angular separation of the two beams was set at 2θ ) 30° creating a 1 µm period pattern when using the 514.5 nm Ar+ laser line. The time evolution of the photoinduced grating was monitored with a resolution of 1 s during a 1500 s exposure (pump on) followed by a 1000 s relaxation period (pump off); the probe beam was either a linearly polarized or a circularly polarized (with a quarter-wave plate) He-Ne light (TEM00, λ ) 632.8 nm, 5 mW) and a silicon diode was used as a detector. Analyses of the transmitted beam (S0) and of the diffracted orders (S+1 and S-1) were made using a Polaroid oriented either parallel or perpendicular to the grating grooves. Theoretical Approach In the following, each polarization configuration is presented separately: In each case, the Jones matrix of the resulting field is first given; then, according to the selective angular holeburning (AHB) model 24,25 and considering a small angle of incidence (θ), the transmission phase matrix of the birefringence grating Jphase is expressed in the zero order and the (1 first-
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TABLE 1: Polarization Modulation of the Interfering Fields for Grating Inscription in Four Distinct Geometries: (p+p) and (s+(-s)), (p+s), and (circular+circular) configurationsa
a The polarization states are given according to the reference axes (X,Y) and are reported as function of the phase difference δ between the two pump beams. The intensity expressions in the (X,Y) plane are given assuming cos θ ) 1.0
order terms in the so-called Tphase matrix. The dephasing induced by the surface relief grating Trelief is then considered and the matricial product (Tphase‚Trelief) gives the resulting Jones matrix. Finally, taking into account for the polarization of the probe beam, theoretical expressions of the (1 diffracted orders with polarization analyses are derived. (p+p) Polarized Pump Beams. For two incident (p) polarized beams, the resulting field in the film plane is modulated along the X direction as shown in Table 1 and its amplitude is equal to 2E2pump cos2 θ cos2(δ/2).
(
cos θ cos(δ/2) Pˆ p + Q ˆ p ) Epumpx2 0 -i sin θ sin (δ/2)
)
(1)
The phase difference between the two incident waves δ is a function of the position X (where X has the direction of the grating vector) and of the grating spacing, Λ; it can be expressed as
δ)
2πX Λ
(2)
In the following, one can assume cos θ equal to 1.0 since we are working with a small incidence angle. Considering the AHB model under low pump conditions, the transmission matrix in the (X,Y) plane describes the polarization on a phase hologram created by both linearly polarized interfering beams and is given by
[Jphase]X,Y ) eiφ0
(
ei∆φ(3cosδ+1) 0 0 ei∆φ(cosδ-1)
)
(3)
For weak birefringence effects (∆φ , 1), (3) becomes:
[Jphase]X,Y ) eiφ0
(
1 + i∆φ(3cosδ + 1) 0 1 +i∆φ(cosδ - 1) 0
)
(4)
where φ0 ) [(-2πd)/λ][(n| + n⊥)/2] is the isotropic part and
∆φ ) [(-2πd)/λ][(n| - n⊥)/2] ) (-πd∆n)/λ is the anisotropic phase shift due to the photoinduced birefringence ∆n, d is the film thickness, and λ is the probe wavelength. Note that the symbols | and ⊥ refer to the X and Y direction, respectively. The phase matrix can be simplified into zero (transmitted) and (1 (diffracted) order terms, respectively:
[Tphase]X,Y ) eiφ0
{(
)
1 + i∆φ 0 + 0 1 - i∆φ i∆φe-iδ 3 0 i∆φeiδ 3 0 + 0 1 2 0 1 2
( )
( )}
(5)
On the other hand, the surface modulation effect d + ∆dcos(δ + φ0) can be expressed by the following Jones’ matrix as
( )
[Trelief] ) ei∆ψ cos(δ+φ0)
1 0 0 1
(6)
where ∆ψ ) (πneff∆d/λ) is directly related to the surface relief modulation, neff is the effective refractive index associated with the inhomogeneous layer (polymer+air), 2∆d is the total surface relief height, and φ0 is the possible dephasing shift between the surface relief and birefringence gratings. It is noteworthy that the above surface relief expression can be developed into modified Bessel functions according to ∞
Ij(i∆ψ)cos[j(δ + φ0)] ∑ j)1
ei∆ψcos(δ+φ0) ) I0(i∆ψ) + 2
(7)
where, according to Olver,26 the modified Bessel function Ij(i∆ψ) is related to the Bessel function Jj(∆ψ) by the relation: Ij(i∆ψ) ) e(iπ/2)j Jj(∆ψ). For a circularly polarized probe light (∩), the Jones’ matrix of the probe is given by S∩ ) E0/x2( 1i ), and the intensity, given by the product of the Jones matrix with its conjugate, may be written as
I ) [Tphase‚Trelief‚S]‚[Tphase‚Trelief‚S]*
(8)
This leads to the expressions of the polarized intensities of the
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diffracted orders (S(1) along the X (I|) and Y (I⊥) direction with respect to the incidence plane, as shown in Figure 1
E20 I|(S(1) ) 2 9 2 2 ∆φ (J0 + J22) + (1 + ∆φ2)J21 + 3∆φ(cos φ0)[J1 (J0 - J2)] 4 9 - 3∆φ2(sin φ0)[J1(J0 + J2)] - ∆φ2J0 J2(cos 2φ0) 2 (9a)
[
]
I⊥(S(1) )
[
E20 2
]
1 2 2 ∆φ (J0 + J22) + (1 + ∆φ2)J21 + ∆φ(cos φ0)[J1 (J0 - J2)] 4 1 (∆φ2(sin φ0)[J1(J0 + J2)] - ∆φ2J0 J2(cos 2φ0) 2 (9b)
where Jn stands for Jn(∆ψ), the Bessel function of order n. (s+(-s)) Polarized Pump Beams. In this case, the intensity of the resulting electric field is modulated only along the Y direction, i.e., perpendicularly to the grating vector (see Table 1):
(
0 x ˆ s ) Epump 2 -i sin(δ/2) Pˆ s + Q 0
)
(10)
Calculation of the induced susceptibilities in the AHB model gives rise to the phase matrix:
[Jphase]X,Y ) eiφ0
(
e-i∆φ(cosδ+1) 0 0 e-i∆φ(3cosδ-1)
)
I⊥(S(1) )
[
E20 2
]
9 2 2 ∆φ (J0 + J22) + (1 + ∆φ2)J21 - 3∆φ(cos φ0)[J1(J0 - J2)] 4 9 (3∆φ2(sin φ0)[J1(J0 + J2)] - ∆φ2J0.J2(cos 2φ0) 2 (14b)
(p+s) Polarized Pump Beams. In the present orthogonal polarization setup, the intensity of the interfering field is constant in the (X,Y) film plane and only changes in the resultant polarizations, between linear, elliptical, and circular states, are effective (Table 1). Indeed, one obtains:
ˆp ) Pˆ s + Q
{(
(
[Jphase]X′,Y′ ) eiφ0
{( )
[Tphase]X,Y ) eiφ0
One can develop (12) into the zero S0 and first S(1 order terms:
[Tphase]X,Y ) e
iφ0
{(
)
1 - i∆φ 0 0 1 + i∆φ i∆φeiδ 1 0 i∆φe-iδ 1 0 0 3 2 0 3 2
( )
( )}
(13)
E20 I|(S(1) ) 2 1 2 2 ∆φ (J0 + J22) + (1 + ∆φ2)J21 - ∆φ(cos φ0)[J1(J0 - J2)] 4 1 -∆φ2(sin φ0)[J1(J0 + J2)] - ∆φ2J0J2(cos 2φ0) 2 (14a)
]
)
(16)
( )}
( )
1 0 i∆φeiδ 0 1 i∆φe-iδ 0 1 + + 0 1 1 0 2 1 0 2 (17)
[
E20 ∆φ2 2 ∆φ2 (J0 + J22) + J21 J J (cos 2φ0) ( 2 4 2 0 2
]
∆φ(sin φ0)[J1(J0 + J2)] (18a) I⊥(S(1) )
[
E20 ∆φ2 2 ∆φ2 (J0 + J22) + J21 J J (cos 2φ0) 2 4 2 0 2
]
∆φ(sin φ0)[J1(J0 + J2)] (18b) or for a linearly polarized probe beam (19a and b),
IVV(S(1) ) E20J21 ≡ IHH(S(1)
When considering a circularly polarized probe, the result of the matricial product leads to the polarized intensities I| and I⊥ for the S(1 orders:
[
(
ei(cos δ)∆φ 0 0 e-i(cos δ)∆φ
Theoretical expressions of the S(1 diffracted orders can thus be derived either for a circularly polarized probe beam (18a and b),
I|(S(1) )
)
(15)
Note that this matrix is diagonal only in a (π/4 rotated (X′,Y′) plane, so that, after transformation into the laboratory or film frame and for weak birefringence values (∆φ , 1), one obtains
(11)
1 - i∆φ(cosδ+1) 0 1 - i∆φ(3cosδ - 1) 0 (12)
)}
The phase matrix and the derived Jones’ anisotropy matrix are similarly derived, assuming θ equal to 0:
and for weak birefringence (∆φ , 1)
[Jphase]X,Y ) eiφ0
)(
cosθsin(δ/2) Epump cosθcos(δ/2) cos(δ/2) - i -sin(δ/2) x2 sinθcos(δ/2) sinθsin(δ/2)
[
IVH(S(1) ) E20
(19a)
]
∆φ2 2 [J + J22 - 2J0J2 (cos 2φ0)] ≡ 4 0 IHV(S(1) (19b)
In the above case, the notation IVV refers to a probe beam and analyzer polarized along the Y direction, while IHH refers to a probe beam and analyzer polarized along the X direction. In all cases, by using the zero order transmitted probe beam (before any grating formation) for normalization, i.e., I0 ) E20 and taking account for the transmission factors of the Polaroid elements, one can measure independently the polarized relative intensities I|(S+1), I⊥(S+1) on the one hand and IVV(S+1), IVH(S+1) on the other one. Similar measurements could be performed for the S-1 diffracted order as well as for the total diffracted intensities Itotal-
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Figure 3. Experimental intensity curves obtained for the S+1 diffraction order as a function of time using (s+(-s)) recording conditions and a circularly polarized probe beam.
Figure 2. (a) Experimental intensity curves obtained for the S+1 first order diffracted beam as a function of time using (p+p) recording conditions and a circularly polarized probe. (b) Zoom of the results obtained at short time.
(S(1) and the equalities Itotal(S(1) ) I|(S(1) + I⊥(S(1) were checked experimentally. Moreover, it must be pointed out that these calculations can be generalized to the higher diffracted orders. Experimental Results (p+p) Configuration. The experimental curves for I|(S+1), I⊥(S+1), and Itotal(S+1) of the first-order diffracted beam in a p(DR1M-co-MMA) grating are shown in Figure 2a during the irradiation and relaxation periods. A zoom of short time results is presented in Figure 2b. When the pump laser is on (after a 60 s delay), the dynamics of the diffraction efficiency of Itotal(S+1) exhibits a fast growth during the first seconds limited to a maximum efficiency of η ) 0.3%. A slower process takes place after 100 s of irradiation and displays a nearly linear variation up to a diffraction efficiency of about 20% after 1400 s. It is noteworthy that similar measurements performed on a larger timescale allow to reach a maximum diffraction efficiency of about 35% after 3 h at 514.5 nm under 90 mW/cm2 irradiance. Surprisingly, one notes that this value is twice that obtained in a (circular+circular) configuration under nearly the same experimental conditions.27 The time dependence behaviors of I|(S+1) and I⊥(S+1), are quite different during the first tens of seconds: the experimental response of I⊥(S+1) is not very sensitive to a fast process and displays a slow linear growth, while that of I|(S+1) varies strongly and is responsible for the dynamics of Itotal(S+1). Then,
after approximately 400 s, the polarized relative intensities of I|(S+1) and I⊥(S+1) behave similarly. When the pump is turned off at t ) 1400 s, there is a slight increase in the diffraction efficiency, probably due to the thermal relaxation in cis to trans isomerization and also to a mechanical relaxation in the three-dimensional organization of the grating; this process could involve some modifications in the molecular reorientations and take origin in the existence of internal pressure gradients. In any way, the efficiency remains constant on a long timescale and all the curves confirm that the surface relief grating is permanent. (s+(-s)) Configuration. The response curves recorded for the diffracted first order with a circularly polarized probe are shown on Figure 3. A maximum diffraction efficiency, which is mainly due to the I⊥ component, is observed in the first one hundred of seconds and the higher diffraction value measured for Itotal(S+1) reaches only η ) 0.4%. After this fast process, the I⊥ response decreases drastically down to a plateau value of nearly negligible efficiency (0.1%), whereas the I| response remains constant but is always very weak. The shape of Itotal is thus governed by the first contribution, so that the total diffraction efficiency is negligible, even weaker than that obtained in related doped polymer systems.21,27 When the pump is turned off, all the curves decrease drastically indicating that the grating is not permanent in this polarization setup and the very weak remaining diffraction effect is surely due to only a small birefringence contribution. (p+s) Configuration. In this linear orthogonal polarization setup, we have performed polarization analyses of the grating formation by using either a circularly (Figure 4a) or a linearly polarized (Figure 4b) probe beam. Similarly to the (s+(-s)) above case, the diffraction efficiency is limited to about 1% maximum, whatever the polarization of the probe. Even though the experimental data points are noisy and scattered, the curves of I|, I⊥, Itotal, and IVH components exhibit a similar shape, with an increase at short time and a plateau value afterthat, and they can be nicely compared to those obtained under (circular+ circular) pump conditions in the doped polymer system.21,27 We note a distinct behavior on the IVV curve which displays a very weak and slightly decreasing response at long time. It must be underlined that the polarized intensities I| and I⊥ are quite similar so that, according to (18a and b) the contribution of the last term proportional to sin φ0 is not significant. Consequently, the dephasing shift between the two competiting processes (i.e., birefringence and surface relief
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Figure 4. Experimental intensity curves obtained for the S+1 diffraction order as a function of time using (p+s) recording conditions and (a) a circularly polarized probe beam and (b) a linearly polarized probe beam.
gratings) must be equal to 0 or (π. One notes also differences in IVH and IVV responses indicating that the birefringence contribution is dominant and the surface relief modulation is probably not significant. Time Variations of the Birefringence and Surface Relief In this section we discuss the strategies used to solve numerically the equation systems established in the theoretical part in order to determine the phase shifts due to the anisotropy ∆φ (birefringence grating) and the surface relief ∆ψ (surface grating). Then, the induced birefringence (∆n) and surface modulation (2∆d) time variations are directly determined knowing the thickness of the sample before grating inscription (d ) 1 µm), the probe wavelength (λ ) 632.8 nm) and the effective index (neff ) 1.25) of the grating.21,27 (p+p) Configuration. As reported in previous studies,4,8 gratings inscribed in (p+p) polarization conditions may reveal a large surface relief modulation with depths from peak to trough nearly equal to the starting flat sample thickness, 1 µm. In this polarization setup, one may assume that the phase shift due to the birefringence grating, ∆φ, is much lower than the phase shift due to the surface relief grating ∆ψ, so that in a first approximation all the terms in ∆φ2 can be neglected in (9a and b). The result is
E20 2 I|(S+1) ≈ [J1 + 3∆φ(cos φ0)[J1(J0 - J2)]] 2
(20a)
E20 2 I⊥(S+1) ≈ [J1 + ∆φ(cos φ0)[J1(J0 - J2)]] 2
(20b)
The second terms in these expressions are probably not significant since both curves in Figure 2a are very similar.
Figure 5. Relevant parameter variations during the grating formation under (p+p) recording conditions: Time evolution of (a) the amplitude of the surface relief modulation (2∆d) and (b) the photoinduced birefringence (-∆n).
However, we note on Figure 2b that at short time I|(S+1) is greater than I⊥(S+1), so that the cos φ0 term must be positive and the dephasing parameter φ0 may be assumed equal to 0. Then, the phase shifts of the birefringence grating (∆φ) and of the surface relief grating (∆ψ) are obtained from these two polarized measurements; their determinations are carried out by solving the following equalities:
∆φ )
2I|(S+1) - J21 3[J0J1 - J1J2]
)
2I⊥(S+1) - J21 [J0J1 - J1J2]
(21)
The procedure consists in determining the best fit values of the phase shift due to the surface relief grating, ∆ψ, which verify the second equality of (21). Then, the phase shift of the birefringence grating, ∆φ, is determined by using the first equality of the above equation. This numerical method requires to pay a particular attention to the incrementation in ∆ψ in order to get a good accuracy in the estimation of both parameters. The time dependences of the surface modulation 2∆d(t) and the birefringence ∆n(t) are reported in Figures 5a and b, respectively. The surface relief increases monotonously and the maximum value of about 350 nm, reached after 1500 s of irradiation, is in perfect agreement with AFM measurements. This confirms that, after the pump beam is turned off, the surface relief remains permanent. Results of the numerical calculations of equality (21) give rise to a positive value for ∆φ, a result in good agreement with the expected negative birefringence if one considers an angular reorientation of the chromophores perpendicularly to the incident
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field, i.e., n| < n⊥ . The maximum value obtained for ∆n is about -0.027 at short time, a value in good agreement with the values observed by Dumont et al.24,25 on thin films of a similar system. In addition, this value compares nicely to those in gratings inscribed using circularly polarized pump lights and estimated by using a similar approach. After few seconds, the birefringence decreases drastically and tends rapidly to zero. This last observation can be explained by the existence of competing processes between the surface relief grating and the phase grating. The dominant process (i.e., the formation of the surface relief modulation) tends to erase completely the birefringence grating. All these results corroborate the validity of the data treatments and of the above used assumptions. Finally, one can expect strong perturbations in the molecular orientations as already demonstrated from micro-Raman confocal spectroscopy measurements.22 (s+(-s)) Configuration. In this experimental setup we have shown that the diffraction efficiency is quite low, so that the phase shifts ∆ψ, ∆φ are surely significantly lower than 1.0. Whatever the dephasing value φ0 equal to 0, (π/2, or π, it is easy to demonstrate that the I⊥(S+1) response is always expected larger than the I|(S+1) one. For simplicity and by analogy with previous results,21 we assume that this relative phase shift between the two processes φ0 is equal to zero, so that (14a and b) may be simplified as
1 ∆φ2 (J0 - J2)2 + J21 - ∆φ(J0J1 - J1J2) + 4 [J21 - 2I|(S+1)] ) 0 (22a)
[
]
9 ∆φ2 (J0 - J2)2 + J21 - 3∆φ(J0J1 - J1J2) + 4 [J21 - 2I⊥(S+1)] ) 0 (22b)
[
]
Numerical simulations of such a mathematical system demonstrate that only one pair of solutions is physically reasonable and it leads to negative values of ∆φ. Indeed, for (s) polarized electric fields one expects a situation with n| > n⊥, so that ∆n must be positive. Calculations are thus performed by varying ∆ψ (the argument of the Bessel functions) until the second equality of (23) is verified:
∆φ )
(J0J1 - J1J2) - x∆1 3(J0J1 - J1J2) - x∆2 ) 1 9 (J0 - J2)2 + 2J21 (J - J2)2 + 2J21 2 2 0
(23)
where ∆1 and ∆2 are the discriminants of (22a and b). Under these conditions, we have extracted the (2∆d) and (∆n) time variations which are reported in Figures 6a and b, respectively. The obtained results confirm the existence of a very weak surface relief modulation (2∆d < 10 nm), even though it is permanent after the pump is turned off. The positive birefringence is also very weak and is partly erased during the surface relief formation. After turning the pump off, this birefringence spontaneously disappears as previously observed in doped polymer systems.21 Finally, it is important to note that, for such gratings of very low efficiency, the diffraction is stongly dependent on the formation of a surface relief, even it is of small amplitude. Confirmations of such phenomena are under investigations by AFM measurements. In any way, it is clear that this process is not at all efficient for surface grating inscription.
Figure 6. Relevant parameter variations during the grating formation under (s+(-s)) recording conditions: Time evolution of (a) the amplitude of the surface relief modulation (2∆d) and (b) the photoinduced birefringence (+∆n).
(p+s) Configuration. In this experimental setup (Scheme 1b), it is known that the interference pattern of two coherent waves with orthogonal linear polarizations has a constant intensity but a polarization state that is periodically modulated. The ligth polarization changes from linear to elliptic, circular, orthogonal elliptic, orthogonal linear, and so on; the directions of the linear polarizations are at (45° with respect to those of the recording waves (see Table 1). According to previous studies,18-20 one thus expects linear anisotropy in the grating regions with linear polarization, both linear and circular anisotropies in those of elliptic polarization and circular anisotropy in regions with circular polarization. We have performed two series of experiments in this case: (i) In the first one, with the probe beam circularly polarized (Figure 4a), we observe quite similar I|(S+1) and I⊥(S+1) dynamical responses demonstrating, according to the theoretical (18a and b) that the dephasing parameter φ0 is equal to zero. (ii) In the second set of experiments, with the probe beam linearly polarized (but in the orthogonal direction with respect to the grating grooves), we observe quite distinct IVV(S+1) and IVH(S+1) dynamical responses. From the IVV(S+1) curve, one can easily get ∆ψ(t) from (19a). Then, ∆φ(t) is obtained from the IVH(S+1) curve, considering the above determined ∆ψ(t) and assuming φ0 ) 0. The final time variations of (2∆d) and of ∆n are thus reported in Figure 7a and b, respectively. As expected, we conclude that a very small, nearly negligible, surface modulation (few nanometers) is generated and then decreases during the irradiation period. However, this could come from the fact that the two interfering beams are not
Azopolymer Holographic Diffraction Gratings
Figure 7. Relevant parameter variations during the grating formation under orthogonal (p+s) polarization recording conditions: Time evolution of (a) the amplitude of the surface relief modulation (2∆d) and (b) the photoinduced birefringence (-∆n).
perfectly orthogonal and give rise to a very weak modulation in the incident intensity. In any way, this effect is not permanent and the phase grating is by far dominant. Indeed, extracted values for the birefringence are in the right order of magnitude for birefringence in such a functionalized glassy polymer system and we observe a plateau value equal to -0.037 at long time. This represents the maximum birefringence value so far obtained over all the experimental configurations, and even though it is not totally permanent when the pumps are turned off, the birefringence effect is clearly the dominant process under this configuration setup. Discussion The main results obtained in this study are summarized in Table 2, in which we have included some results of our previous measurements using two circularly polarized interfering beams on the same functionalized copolymer system.21 In the (p+p) configuration, the resulting electric field is polarized along the X direction and the birefringence is positive as expected; in contrast, in the (s+(-s)) configuration, the birefringence is negative since the electric field is polarized along the Y direction. These observations are in agreement with previous polarimetric measurements25 in which it was demonstrated that the refractive index along the direction of the electric field polarization was lower than those in the orthogonal directions. Nevertheless, it is surprising to observe nearly the same order of magnitude in the (p+p) and (co-+contra-circular) polarization conditions,
J. Phys. Chem. B, Vol. 103, No. 32, 1999 6697 since in the last case the total incident electric field in the film plane is linearly polarized with different directions in the various areas of the grating and is never equal to zero; when using two linearly (p) polarized beams, the photoinduced orientation of chromophore and birefringence would be expected weaker, since alternated regions of high and low rate of isomerization are formed. In fact, from polarized micro-Raman confocal measurements we have recently shown that the orientation in the different areas of a grating inscribed using two circularly polarized pump beams was strongly perturbed particularly in the pit regions;22 this perturbation probably comes from a modification in the chromophore orientation due to large pressure gradients and to a viscoelastic flow of the polymer chains as first proposed by Rochon et al.7-9 Similar microRaman experiments on gratings inscribed in the (p+p) configuration should reveal similar or even larger effects in the chromophore orientations because of the significant surface relief modulation. Furthermore, the formation of the surface relief, which is largely governed by the light intensity modulation along the grating X direction, can also explain the large differences in birefringence observed for (p+p) and (s+(-s)) configurations. Actually, it must be recalled that in the present copolymer system under study two (s) polarized beams do not produce any surface modulation, two circular polarized beams produce gratings of moderate efficiency (10-15%) and two interfering (p) polarized beams give rise to the greatest magnitude and rate of surface modification, with a first-order diffraction efficiency increasing up to 20-25%. Such results are in contrast with data reported on other polymer systems by several groups4,12,17 who have claimed that the (circular+circular) setup was the most efficient. One could invoke different bulk polymer properties but, in our opinion, these discrepancies can be rationalized by a careful inspection of the dynamic processes in both configurations. In the (circular + circular) setup we have shown21 that the birefringence and surface relief gratings were still competing during the 1500 s irradiation period, whereas in the present (p+p) study it is clear that the time dependence efficiency is essentially governed by the surface relief modulation. So, the hierarchy in efficiency appears to be very dependent on the pump intensity and on the irradiation time. Indeed, with the copolymer system under study we have observed that, using similar pump irradiances, the (circular+circular) and (p+p) setup were the most efficient at short time and at long time, respectively and the crossing point of their diffraction efficiency curves was varying strongly upon the experimental conditions.28 This emphasizes the usefulness of our time dependent analyses in order to establish the real-time variations of the birefringence and of the surface relief. Under these conditions, all reported results can be reconciled but this raises a question still open to discussion: why is the (circular+circular) pump condition generally more efficient in other polymer systems? We think that this is likely due to larger birefringence contributions and to higher rates of isomerization in the photostationnary state. Concerning the (p+s) polarization condition it is remarkable that, even though the grating is of very weak efficiency (1%), the total birefringence induced by linear, elliptic and circular polarization effects has the greatest magnitude; this is an important result which can be correlated with recently reported anomalous diffraction phenomena, in particular the formation a half-period grating, in a high birefringent liquid-crystal type azopolymer system.18-20 In that case the above theoretical approach cannot be directly applied but must be generalized
6698 J. Phys. Chem. B, Vol. 103, No. 32, 1999
Lagugne´ Labarthet et al.
TABLE 2. Summary of Maximum Diffraction Efficiency (η%), Birefringence (∆n), and Surface Relief Amplitude (2∆d) for Gratings Inscribed on Films of the Copolymer p(DR1M-co-MMA) under Various Polarization Configurations and Using an Irradiance of 90 mW/cm2 (λ ) 514.5 nm). Data Reported for the (circular+circular) Setup Come from Ref 21 under irradiation polarization configurations
maximum efficiency η% (S+1)
maximum birefringence (∆n)
maximum surface relief (2∆d) nm
(p+p) (s+(-s)) (p+s) (circ.+circ.)
20-25 0.1 1.0 10-15
-0.028 +0.007 -0.037 -0.032
350
