Mueller imaging polarimetry of holographic polarization gratings inscribed in azopolymer films

Geminiano Martinez-Ponce

doi:10.1364/OE.24.021364

1. Introduction

For a long time, photoreactive films made of side-chain azobenzene-containing polymers (SCAPs) have been used as an efficient holographic media for the recording of polarization gratings (PGs) [1]. These diffractive elements are recorded by superimposing two mutually coherent light beams having orthogonal polarization states on the light-responsive film, e. g., left and right circular polarization [2]. The recording wavelength λ is located within the absorption band π → π^* of the photoreactive medium. Propagation vectors subtend an small angle but the resultant light field does not show any interference fringes. However, the polarization state of the light field is spatially modulated and it induces anisotropic changes in the optical properties of the SCAP film. Under certain conditions, a surface relief grating is formed because there is a pressure force proportional to the gradient of the electric field on the film surface that produces a mass transport [3–5]. In the most of reported azopolymers, the recorded gratings are stable for a long time.

Optical anisotropy recording is possible because azobenzene moities are responsive to the electric field of the excitation beam. When a linearly polarized light propagates through an SCAP film, the reactive moieties experience, after several trans-cis-trans isomerizations, an antialignment of the molecular axis which is reflected macroscopically as optical linear anistropies, namely, linear diattenuation and birefringence [6–8]. On the other hand, supramolecular chiral structures are induced when an elliptically polarized beam of light propagates through the photoresponsive material showing, as a consequence, optical circular anisotropies (mainly, optical rotation) [9–12]. Multiple writing/erasure cycles has been reached by using alternately linear and circular polarized light [13] and also two different excitation wavelengths [14].

One of the features observed in PGs photoinduced in SCAPs is the decomposition of incident polarized light in basic polarization states, property that has been proposed to develop polarimetric instruments [15–18]. On the other hand, it has been also reported that specific spatial distribution of photoinduced optical anisotropies generate, by diffraction, complex beams [19,20]. Thus, SCAP films act as spatial light modulators to design optical tweezers. In the analysis of the diffraction efficiency dynamics, it has been assumed that two kinds of phase gratings are recorded, one related to the bulk refractive index modulation and the other to the surface relief modulation [21–23].

In this paper, it is presented a theoretical and experimental polarimetric analysis, within the frame of Mueller-Stokes formalism, of different configurations traditionally used to form polarization gratings in a film made of an azobenzene-containing polymer. In order to evaluate the optical anisotropies distribution in the bulk of the film, a Mueller polarimetric microscope based on rotating waveplates has been used. The 16 images in the Mueller matrix are then processed by computer using the polar decomposition with the aim to calculate the maps of diattenuation, depolarization, linear retardance and optical rotation. The basic anisotropic optical properties shown separately will provide a better understanding of the recording process in these kind of photoanisotropic materials.

2. Theory

A holographic polarization grating is recorded when two electromagnetic plane waves, both having wavelength λ and being mutually coherent, are superimposed upon a light-responsive film. A requirement to improve the recording of vectorial anisotropy, if the optical beams propagate collinearly (k₁ = k₂), states that their electric field vectors E₁(r, t) and E₂(r, t) should be orthogonal; thus, E₁(r, t) · E₂(r, t) = 0, where r = xî + yĵ + zk̂ is the position vector. However, in order to obtain a spatially modulated polarized field, the propagation vectors k₁ and k₂ ought to subtend a small angle 2θ which is bisected by the normal n̂ to the film surface. Even though the spatial frequency depends directly on θ, a large angle affects the polarization pattern because interference fringes are produced. Let us assume that propagation and normal vectors lie in the incidence plane xz. The resultant electric field distribution is given, mathematically, by the real part of the addition of the complex electric field amplitudes [24],

\begin{array}{l} ℜ {Σ (r, t)} & = ℜ {E_{1} (r, t) + E_{2} (r, t)}, \\ = ℜ {[E_{01} e^{j k_{1} \cdot r} + E_{02} e^{j k_{2} \cdot r}] e^{- j ω t}}, \\ = p (x, z) cos ω t + q (x, z) sin ω t, \end{array}

where k₁ · r = k(−x sin θ + z cos θ) and k₂ · r = k(x sin θ + z cos θ), k = 2π/λ and

E_{01} = E_{01} (cos ψ_{2} cos θ i + sin ψ_{2} e^{- j ϕ_{1}} j + cos ψ_{2} sin θ k),

E_{02} = E_{02} (cos ψ_{2} cos θ i + sin ψ_{2} e^{j ϕ_{2}} j - cos ψ_{2} sin θ k) .

ψ₁ (ψ₂) is the angle between the incidence plane and the electric field vector E₀₁ (E₀₂), ϕ₁ and ϕ₂ are the phase differences between the y-axis component and the in-plane xz component for E₀₁ and E₀₂, respectively. Eq. (1) is the parametric representation of an ellipse where the length L_a,b of the semiaxes are given by [25]

L_{a, b}^{2} = \frac{1}{2} [p^{2} + q^{2} \pm \sqrt{{(p^{2} - q^{2})}^{2} - 4 {(p \cdot q)}^{2}}],

where p = |p| and q = |q|. Direction cosines (l_x, l_y, l_z) = (cos α_x, cos α_y, cos α_z) of the semi-major axis can be obtained from

\pm l_{i} = {[\frac{1}{1 + ξ}]}^{1 / 2} \frac{p_{i}}{L_{a}} + {[\frac{ξ}{1 + ξ}]}^{1 / 2} \frac{q_{i}}{L_{a}}, i = x, y, z,

where L_a is the length of the major semiaxis and

ξ = \frac{L_{a}^{2} - q^{2}}{L_{a}^{2} - p^{2}} .

Descriptors of polarized light, ellipticity e and orientation α of the polarization ellipse with respect to x-axis, are given by

e = \frac{L_{b}}{L_{a}},

α = α_{x} .

Irradiance distribution at the superposition plane, considering E₀₁ = E₀₂ = E₀, is given by

I (x, y) = 2 I_{0} [1 + cos ψ_{1} cos ψ_{2} cos 2 θ cos (2 k x sin θ) + sin ψ_{1} sin ψ_{2} cos (2 k x sin θ - 2 ϕ)],

where

I_{0} = ε_{0} c E_{0}^{2} / 2

, and ϕ = ϕ₂ − ϕ₁ is the absolute phase difference. It can be noticed that I(x, y) is constant solely when ψ₁ = 0 and ψ₂ = π/2 or conversely. Consequently, under this conditions, there is pure polarization modulation [26]. Any other polarization setting will produce an irradiance modulation, i. e., an interference pattern with spatial frequency f_x = 2 sin θ/λ that may induce a change proportional to the local intensity in the average refractive index. In the following, the equations to describe the modulated polarization electric field and the Mueller matrices to model the photoinduced anisotropy recorded in the film are derived.

2.1. Case I: ψ₁ = 0, ψ₂ = π/2 and ϕ = 0

This recording conditions corresponds to both linearly polarized beams E₁ and E₂, but one is oriented parallel and the other perpendicular to the incidence plane. For the sake of simplicity, let us consider the modulation over the plane z = 0. Then,

p (x, z = 0) = {E_{01} cos θ i + E_{02} j + E_{01} sin θ k} cos [k (x sin θ)],

q (x, z = 0) = {- E_{01} cos θ i + E_{02} j - E_{01} sin θ k} sin [k (x sin θ)] .

The plane that contains the polarization ellipse is not parallel to the superposition plane because the parametric vectors have a z-component. In order to reproduce the pattern shown in Fig. 1(a) using Eq. (1), a further simplification should be applied to Eqs. (9). It consists in assuming that E₀₁ = E₀₂ = 1, sin θ ≈ 0 and cos θ ≈ 1. However, the phase modulation term kx sin θ is not affected by the small angle approximation. Thus,

ℜ {Σ (x, t)} = (i + j) cos (k x sin θ) cos ω t - (i - j) sin (k x sin θ) sin ω t .

Descriptors of polarization ellipse are obtained from Eqs. (3) and (4), as

L_{a}^{2} = 1 \pm cos (2 k x sin θ)

,

L_{b}^{2} = 1 \mp cos (2 k x sin θ)

, and α_x = ±π/4. Then, ellipticity e = L_b/L_a is spatially modulated and the period Λ_x = λ/ (2 sin θ) is the same observed in the interference pattern.

Fig. 1 Result of coherent superposition in typical configurations for the recording of polarization gratings in photoanisotropic materials.

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The spatial distribution of the electric field described by Eq. (10) can be expressed in an alternative notation using the Stokes vector S = {I, Q, U, V}^T, where T is for the transpose. Recalling that

I = 〈 Σ_{x} Σ_{x}^{*} 〉 + 〈 Σ_{y} Σ_{y}^{*} 〉,

Q = 〈 Σ_{x} Σ_{x}^{*} 〉 - 〈 Σ_{y} Σ_{y}^{*} 〉,

U = 〈 Σ_{x} Σ_{y}^{*} 〉 + 〈 Σ_{y} Σ_{x}^{*} 〉,

V = j (〈 Σ_{x} Σ_{y}^{*} 〉 - 〈 Σ_{y} Σ_{x}^{*} 〉),

where Σ(x, t) = Σ_xi + Σ_yj,

j = \sqrt{- 1}

, * and 〈〉 are for complex conjugate and time average, respectively. Thereby,

S = {1, 0, cos (2 k x sin θ), sin (2 k x sin θ)}^{T} .

Parameters U and V are modulated in quadrature along x-axis and their spatial periods are also Λ_x. The photoanisotropic film is responsive to this electric field modulation and changes are induced in its optical properties, such as diattenuation (commonly referenced as dichroism) and retardance. Presupposing that anisotropic response of the material is linearly proportional to the Stokes parameters, the diattenuation vector ^UD is found to be (see Appendix)

{}^{U}D = [\begin{matrix} 0 \\ tanh (U k Δ α_{0}^{l} d) \\ tanh (V k Δ α_{0}^{c} d) \end{matrix}] .

Δ α_{0}^{l}

and

Δ α_{0}^{c}

are the linear and circular vectorial anisotropic extinction constants for the material. The film thickness was defined as d = d₀ + Δd cos(2kx sin θ + δ), where d₀ is the average thickness, 2Δd is the surface relief depth, and δ = π is the phase shift between the anisotropic bulk and isotropic surface modulations [27]. Then, assuming that the polarizance vector P = ^UD, the complete diattenuation Mueller matrix have the form [28]

{}^{U}{M_{D}} = [\begin{matrix} 1 & {}^{U}{D^{T}} \\ P & m_{D} \end{matrix}],

where the diattenuation submatrix is written as

\begin{array}{l} m_{D} & = a I_{3} + b ({}^{U}D \times {}^{U}{D^{T}}) \\ = [\begin{matrix} a & 0 & 0 \\ 0 & a + b {tanh}^{2} (U k Δ α_{0}^{l} d) & tanh (U k Δ α_{0}^{l} d) tanh (V k Δ α_{0}^{c} d) \\ 0 & tanh (U k Δ α_{0}^{l} d) tanh (V k Δ α_{0}^{c} d) & a + b {tanh}^{2} (V k Δ α_{0}^{c} d) \end{matrix}] \end{array} .

In Eq. (15),

a = \sqrt{1 - {| {}^{U}D |}^{2}},

b = \frac{1 - a}{{| {}^{U}D |}^{2}} .

On the other hand, linear and circular birefringence values are also proportional to the corresponding Stokes parameter. The retardance matrix can be written as

{}^{U}{M_{R}} = [\begin{matrix} 1 & 0^{T} \\ 0 & m_{R} \end{matrix}],

where m_R is a pure retardance submatrix given by

m_{R} = [\begin{matrix} cos (U k Δ n_{0}^{l} d) cos (V k Δ n_{0}^{c} d) & cos (U k Δ n_{0}^{l} d) sin (V k Δ n_{0}^{c} d) & - sin (U k Δ n_{0}^{l} d) \\ - sin (V k Δ n_{0}^{c} d) & cos (V k Δ n_{0}^{c} d) & 0 \\ sin (U k Δ n_{0}^{l} d) cos (V k Δ n_{0}^{c} d) & sin (U k Δ n_{0}^{l} d) sin (V k Δ n_{0}^{c} d) & cos (U k Δ n_{0}^{l} d) \end{matrix}] .

Δ n_{0}^{l}

and

Δ n_{0}^{c}

are the linear and circular vectorial anisotropic birefringence constants for the material.

Assuming a homogeneous nondepolarizing medium, the photo-induced linear anisotropy with axis oriented at ±π/4, related to the Stokes parameter U, in the light responsive film can be expressed as a Mueller matrix given by

\begin{array}{l} M_{L} & = {}^{U}{M_{R}} (Δ n_{0}^{c} = 0) {}^{U}{M_{D}} (Δ α_{0}^{c} = 0) \\ = e^{- k α_{e}^{l} d} [\begin{matrix} cosh (k Δ α^{l} d) & 0 & - sinh (k Δ α^{l} d) & 0 \\ 0 & cos [k Δ n^{l} d] & 0 & sin [k Δ n^{l} d] \\ - sinh [k Δ α^{l} d] & 0 & cosh [k α^{l} d] & 0 \\ 0 & sin [k Δ n^{l} d] & 0 & cos (k Δ n^{l} d) \end{matrix}], \end{array}

where

α_{e}^{l}

is the average linear extinction coefficient,

Δ α^{l} = Δ α_{0}^{l} U = Δ α_{0}^{l} cos (2 k x sin θ)

and

Δ n^{l} = Δ n_{0}^{l} U = Δ n_{0}^{l} cos (2 k x sin θ)

are the modulated linear diattenuation and birefringence, respectively.

On the other hand, the Mueller matrix describing the photo-induced gyrotropy in the same film related to the Stokes parameter V is given by

\begin{array}{l} M_{c} & = {}^{U}{M_{R}} (Δ n_{0}^{l} = 0) {}^{U}{M_{D}} (Δ α_{0}^{l} = 0) \\ = e^{- k α_{e}^{c} d} [\begin{matrix} cosh (k Δ α^{c} d) & 0 & 0 & sinh (k Δ α^{c} d) \\ 0 & cos [2 k Δ n^{c} d] & sin [2 k Δ n^{c} d] & 0 \\ 0 & - sin [2 k Δ n^{c} d] & cos [2 k Δ n^{c} d] & 0 \\ sinh (k Δ α^{c} d) & 0 & 0 & cosh (k Δ α^{c} d) \end{matrix}], \end{array}

where

α_{e}^{c}

is the average circular extinction coefficient,

Δ α^{c} = Δ α_{0}^{c} V = Δ α_{0}^{c} sin (2 k x sin θ)

and

Δ n^{c} = Δ n_{0}^{c} Q = Δ n_{0}^{c} sin (2 k x sin θ)

depict the spatial modulation of circular diattenuation and birefingence. The overall Mueller matrix for the polarization grating inscribed in the photo-reactive film is, in this case,

M_{I} = M_{L} M_{c} .

The matrix order for the product is commutable.

2.2. Case II: ψ₁ = −ψ₂ = π/4 and ϕ = 0

In this polarization setting, the two light beams are once again linearly polarized, but the angle subtended between the electric fields and the incidence plane are ±π/4. Then, the polarization state modulation over the plane z = 0 can be traced using the following parametric vectors,

p (x, z = 0) = \frac{\sqrt{2}}{2} {(E_{01} + E_{02}) cos θ i + (E_{01} - E_{02}) j + (E_{01} - E_{02}) sin θ k} cos [k (x sin θ)],

q (x, z = 0) = \frac{\sqrt{2}}{2} {(E_{02} - E_{01}) cos θ i - (E_{01} + E_{02}) j - (E_{01} + E_{02}) sin θ k} sin [k (x sin θ)] .

After applying the small angle approximation, Eqs. (23) are substituted in Eq. (1), which is rewritten as

ℜ {Σ (x, t)} = \sqrt{2} [cos (k x sin θ) cos ω t i - sin (k x sin θ) sin ω t j] .

Evaluation of Eq. (24) plots the modulation pattern in Fig. 1(b) along the x-axis. The length of the semi-major axis of the polarization ellipse is

L_{a}^{1} = 1 \pm cos (2 k x sin θ)

and the orientation angle α_x = π/2 or 0.

The spatial modulation of the Stokes parameters over the superposition plane is

S = {1, cos (2 k x sin θ), 0, sin (2 k x sin θ)}^{T} .

When the SCAP film is exposed to this polarized optical field, linear and circular anisotropies are photo-induced periodically. Diattenuation vector under this condition is given by

{}^{Q}D = [\begin{matrix} tan (Q k Δ α_{0}^{l} d) \\ 0 \\ tan (V k Δ α_{0}^{c} d) \end{matrix}] .

Then, the diattenuation Mueller matrix is

{}^{Q}{M_{D}} = [\begin{matrix} 1 & {}^{Q}{D^{T}} \\ {}^{Q}D & m_{D} \end{matrix}],

where

m_{D} = [\begin{matrix} a + b {tanh}^{2} (Q k Δ α_{0}^{l} d) & 0 & tanh (Q k Δ α_{0}^{l} d) tanh (V k Δ α_{0}^{c} d) \\ 0 & a & 0 \\ tanh (Q k Δ α_{0}^{l} d) tanh (V k Δ α_{0}^{c} d) & 0 & a + b {tanh}^{2} (V k Δ α_{0}^{c} d) \end{matrix}] .

Moreover, the retardance Mueller matrix takes the form

{}^{Q}{M_{R}} = [\begin{matrix} 1 & 0^{T} \\ 0 & m_{R} \end{matrix}],

where m_R is now

m_{R} = [\begin{matrix} cos (V k Δ n_{0}^{c} d) & sin (V k Δ n_{0}^{c} d) & 0 \\ - cos (U k Δ n_{0}^{l} d) sin (V k Δ n_{0}^{c} d) & cos (U k Δ n_{0}^{l} d) cos (V k Δ n_{0}^{c} d) & sin (U k Δ n_{0}^{l} d) \\ sin (U k Δ n_{0}^{l} d) sin (V k Δ n_{0}^{c} d) & - sin (U k Δ n_{0}^{l} d) cos (V k Δ n_{0}^{c} d) & cos (U k Δ n_{0}^{l} d) \end{matrix}] .

The Mueller matrix to depict linear birefringence and diattenuation related to the Stokes parameter Q, with axis oriented at 0 or π/2, is expressed as

\begin{array}{l} M_{l} & = {}^{Q}{M_{R}} (Δ n_{0}^{c} = 0) {}^{Q}{M_{D}} (Δ α_{0}^{c}) \\ = e^{- k α_{e}^{l} d} [\begin{matrix} cosh (k Δ α^{l} d) & - sinh (k Δ α^{l} d) & 0 & 0 \\ - sinh (k Δ α^{l} d) & cosh (k Δ α^{l} d) & 0 & 0 \\ 0 & 0 & cos [k Δ n^{l} d] & sin [k Δ n^{l} d] \\ 0 & 0 & - sin [k Δ n^{l} d] & cos [k Δ n^{l} d] \end{matrix}], \end{array}

where

Δ α^{l} = Δ α_{0}^{l} Q = Δ α_{0}^{l} cos (2 k x sin θ)

and

Δ n^{l} = Δ n_{0}^{l} Q = Δ n_{0}^{l} cos (2 k x sin θ)

are the modulated linear diattenuation and birefringence, respectively.

The Mueller matrix representing the photo-induced gyrotropy in the film is the same as in the Case I, Eq. (21). Then, the compound Mueller matrix for this polarization holographic grating is

M_{II} = M_{l} M_{c} .

2.3. Case III: ψ₁ = ψ₂ = ψ and ϕ = π/2

The last superposition configuration appertains to left and right circularly polarized beams. Consider that ψ = π/4, so the vectors involved in the parametric Eq. (1) are given, in plane z = 0, as

p (x, z = 0) = \frac{\sqrt{2}}{2} {(E_{01} + E_{02}) cos θ cos [k (x sin θ)] i - (E_{01} + E_{02}) sin [k (x sin θ)] j + (E_{01} - E_{02}) sin θ cos [k (x sin θ)] k},

q (x, z = 0) = \frac{\sqrt{2}}{2} {- (E_{01} - E_{02}) cos θ sin [k (x sin θ)] i - (E_{01} - E_{02}) cos [k (x sin θ)] j - (E_{01} + E_{02}) sin θ sin [k (x sin θ)] k} .

Complex field amplitude distribution is given, considering small θ and E₀₁ = E₀₂ = 1, by

ℜ {Σ (x, t)} = \sqrt{2} [cos (k x sin θ) cos ω t i - sin (k x sin θ) sin (ω t + \frac{π}{2}) j] .

The polarization pattern formed along the x-axis is shown in Fig. 1(c). The semi-major axis is constant,

L_{a} = \sqrt{2}

and the orientation angle α_x = kx sin θ.

The Stokes vector describing the field is given by

S = {1, cos (2 k x sin θ), - sin (2 k x sin θ), 0}^{T} .

In consequence, the Mueller matrix for the polarization grating is

M_{III} = M_{L} M_{l} or M_{III} = M_{l} M_{L} .

Note that for M_L the photoanisotropic modulation is given as

Δ α^{l^{'}} = Δ α_{0}^{l} sin (2 k x sin θ)

and

Δ n^{l^{'}} = Δ n_{0}^{l} sin (2 k x sin θ)

. A numerical simulation of Eq. (36) and the experimental measurement of the Mueller matrix composed by images, both produced by the photoanisotropic recording, are shown in Fig. 2. Elements off the main and secondary diagonals show similar features related to linear diattenuation and birefringence. The main diagonal images in the simulation seem to have a higher modulation but their amplitude is close to one. On the other hand, the images in the secondary diagonal show a similar modulation but around one. This appearance is caused because of the hyperbolic functions in the diattenuation matrices.

Fig. 2 a) Numerical evaluation and b) experimental measurement of Mueller matrix images given by the photoanisotropic modulation in the film induced by the superposition of left and right circularly polarized beams. Simulation has been performed using Δn^l = 0.0075, Δα^l = 0.0025, Δd = 100 nm, and d₀ = 4 μm.

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3. Materials and methods

The synthesis of the liquid-crystalline SCAP has been reported elsewhere [29]. The homopolymer was made by radical-type polymerization of an acrylate monomer containing the dye 4-[4-(6-hydroxy-hexyloxy)phenylazo]benzonitrile. The polymer was dissolved at room temperature in chloroform and deposited by casting on glass slides to form 4 μm-thick SCAP films. After solvent evaporation, the polarization gratings were photoinduced in the films by exposition to a light field modulated in polarization. The periodic pattern was generated using the coherent superposition of the three traditional pairs of orthogonally polarized beams described above.

The equal intensity recording beams were obtained by splitting the radiation from a DPSS laser source emitting at λ_r = (473 ± 1) nm and average power ≤100 mW. Power density was set to 250 mW/cm² on the superposition plane using a neutral filter near the laser aperture. The orthogonal polarization states were set in each beam by propagation through a Berek’s compensator. The subtended angle by the recording beams (2θ ≈ 3.3°) produce a modulated pattern with a spatial period Λ_x ≈ 8 μm. The recorded spot diameter was D = 2 mm. The exposure lasts 10 min in each case, which is an approximated photoalignment saturation time. The Mueller imaging polarimeter comprises two fixed dichroic linear polarizers (extinction ratio 10,000:1), two rotating zero-order quarter-wave retarders (design wavelength λ₀ = 633 nm), a He-Ne laser as light source (λ_r = 633 nm), a 60× microscope objective, an achromatic doublet lens as the ocular and a monochromatic CCD camera placed at the optical system output as shown in Fig. 3. The camera resolution was set to 640×480 pixels and the frame rate has been set at 30 frames/s using a 16-bit data format. The image box (1,800 frames) was Fourier analyzed pixel by pixel in order to determine the Mueller matrix. First, in the imaging process, a broadband incoherent light source illuminated the grating surface obliquely. Then, the polarization analyzing microscope was adjusted to image the grating surface on the CCD. This step should be taken in order to avoid the Talbot effect produced when coherent light is diffracted by the grating. The first self-imaging distance using 633nm He-Ne laser illumination is, approximately, 0.2mm while the focal length of the microscope objective using a tube length of 16cm is around 2 or 3mm.

Fig. 3 Mueller imaging polarimeter based on rotating compensators.

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4. Results

4.1. Case I: Horizontal and vertical linear polarizations

The first coherent superposition produces, see Eq. (10) and Fig. 1(a), a light field with the polarization ellipticity spatially modulated, running from a linear to a circular polarization state and backwards keeping the semi-major and -minor axes oriented along 0 and π/2, or viceversa. Linear birefringence and optical rotation induced where light is linearly and circularly polarized, respectively, is assumed to be homogeneous through the film thickness.

After the film sample was exposed, the photorecorded polarization grating was analyzed immediately in the Mueller imaging polarimeter. Anistropy and depolarization maps obtained by polar decomposition of Mueller images are shown in Fig. 5. The map of optical anisotropies corresponding to linear birefringence and diattenuation, as well as the depolarization, are modulated with a spatial frequency which is twice the expected one. It has been reported that the grating spatial frequency decreases half the period because the incident recording beams fulfill the Raman-Nath condition and multiple diffraction orders are generated whilst the grating is recorded [30]. These diffracted beams affect the spatial modulation of polarization ellipse in the recording region. In order to estimate the change in the polarization pattern, let us consider only the ±1 diffraction orders generated by each recording beam. Thus, E₂ is the superposition of one recording beam and the −1 diffracted order generated by the other recording beam. Both beams have the same state of polarization. Similarly, E₁ is the superposition of the second recording beam and the +1 diffracted order generated by the first recording beam, the latter is the reconstruction of the second recording beam. Departing from Eq. (1), it is found that the electric field resulting from the superposition of four linearly polarized propagating beams, two vertically and two horizontally oriented, is given by

\begin{array}{l} Σ (x, t) = (i + j) [cos (k x sin θ) + cos (k x sin 3 θ)] cos ω t \\ - (i - j) [sin (k x sin θ) + sin (k x sin 3 θ)] sin ω t . \end{array}

The polarization pattern given by numerical evaluation of Eq. (37) is shown in Fig. 4. It has been assumed that the ±1 diffraction orders subtend a ±3θ angle with the normal to the photoreactive film an these angles fulfill also the small angle approximation. Based on these considerations, all of the optical anisotropies should be modulated twice the expected spatial frequency because of the recording configuration. In addition, because the electric field component along the film surface exerts pressure on the surface to induce a relief grating, the film thickness should also have half the expected period. The spatial shift between the surface relief and the anisotropic phase modulation has been considered in the previous numerical simulation of Mueller images.

Fig. 4 Polarization pattern formed when recording and diffracted beams are superimposed coherently.

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Fig. 5 Images obtained by polar decomposition of the images obtained by Mueller imaging. a) Diattenuation, b) Depolarization, c) Retardance, and d) Optical rotation.

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The images provided by single value decomposition of Mueller images have information of the product between optical anisotropy and film thickness. Then, as it is observed for linear retardance and diattenuation, spatial modulation increased. Nevertheless, optical rotation is modulated at the fundamental spatial frequency and is not affected by the varying optical path provided by the surface relief modulation. This result does not agree with the previous interpretation.

However, Stokes vector is given by

S = \frac{1}{4} [\begin{matrix} 1 \\ 0 \\ - sin (2 k x sin θ) - 2 sin (4 k x sin θ) - sin (6 k x sin θ) \\ cos (2 k x sin θ) + 2 cos (4 k x sin θ) + cos (6 k x sin θ) \end{matrix}],

where the all the electric field amplitudes have been considered equal and S has been normalized. In Fig. 6, modulation of linear polarization oriented along ±π/4 (U) shows an additional cycle with less amplitude. For long exposure, the photoalignment induced in the film by this linear polarization distribution causes the frequency doubling. On the other hand, Stokes parameter V shows a change in handedness between two minima, but the amplitude is low and the average amplitude is nearly zero. In addition, if diffraction efficiency is not high, the change in handedness is not observed. Then, circular anisotropy will have the fundamental frequency, as shown in Fig 5(d). Depolarization power measures in Fig. 5(b) showed also twice the frequency because there is a region where Q, U, and V are zero between maxima. Then, because there is apparently depolarized light in that region, there is not linear or circular photoalignment.

Fig. 6 Spatial modulation of Stokes parameters U and V when two light beams, s- and p-polarized, are superimposed.

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4.2. Case II: Linear polarizations oriented at ±π/4

In this recording configuration, the light field is spatially modulated in a similar way than in previous case, but the semi-major axis is oriented at α_x = ±π/4. Then, a modulated linear birefringence and optical rotation is also expected to be photoinduced in the SCAP film. In fact, the spatial distribution of the Stokes vector in this recording conditions can be obtained applying a transformation rotation to the previous spatial distribution of S. However, both cases are presented separately because the material response in each case is different.

In Fig. 7, the polarimetric images obtained with the Mueller imaging system show the anisotropy maps recorded in SCAP films. In contrast with the previous case, recording self-diffracted first orders do not interfere decreasing the spatial period of the polarization and surface relief grating. It is clear that there exists a slight linear diattenuation modulated in the same manner than the linear birefringence, which is stronger. Optical rotation is negative and is displaced one fourth of a period with respect to the linear birefringence map. Finally, it can be seen also a depolarization pattern coincident with the optical rotation map. This measure is produced because the azobenzene moities are still in the isomerization process in those places where the electric field does not have a defined orientation (circular polarization). All of the anisotropy and depolarization patterns show the same spatial frequency.

Fig. 7 Same as in Fig. 5 but using the second setup.

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4.3. Case III: Right and left circular polarization

A polarized electric field consisting of linearly polarized light with spatially modulated orientation is obtained as a result of this coherent superposition. Then, only linear birefringence and diattenuation are expected to be photoinduced in the SCAP film. In Fig. 8, polarimetric images of optical anisotropies induced in the film by the polarization field are shown. As expected, linear birefringence and diattenuation maps are in phase and show a spatial modulation given by the incident recording beams. There is not optical rotation and depolarization degree is smaller than in previous cases. It may be connected to the mobility of azobenzene moieties that decreases because there are not points showing circular polarization over the polarized light field.

Fig. 8 Same as in Fig. 5 but using the third setup.

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5. Conclusion

Polarization gratings recorded in thin films made of side-chain azobenzene-containing polymer have been measured using a polarimetric imaging technique. For the first time, to the best of our knowledge, Mueller matrix and its subsequent polar decomposition, which provides basic optical anisotropies, have been used to describe theoretically and experimentally the recording process of this kind of diffractive elements. It was shown that maps of linear (circular) photo-induced optical anisotropies are related to the spatial modulation of the Stokes parameters Q or U (V) in all of the recording settings. In the specific recording case when linearly polarized beams oriented vertically and horizontally (s- and p-polarization, respectively), the modulation of optical rotation showed the fundamental spatial frequency, but linear anisotropies doubled that value. This discrepancy is explained on the basis of V spatial modulation, which resulted from the superposition of the two incident recording beams and the first two (±1) diffracted orders. In conclusion, Mueller imaging polarimetry has been used successfully in the characterization of light-responsive materials.

Appendix

Linear diattenuator with axis along x– and y–axis and thickness d is represented in Jones matrix formalism as

J_{LD} = [\begin{matrix} e^{- k α_{x} d} & 0 \\ 0 & e^{- k α_{y} d} \end{matrix}] = e^{- k \bar{α} d} [\begin{matrix} e^{k Δ α^{l} d / 2} & 0 \\ 0 & e^{- k Δ α^{l} d / 2} \end{matrix}],

where α_i is the attenuation coefficient along i direction (i = x, y), isotropic attenuation coefficient ᾱ = (α_x + α_y)/2, and diattenuation coefficient Δα^l = α_y − α_x. When linear diattenuator is rotated π/4, Jones matrix is

J_{{LD}^{'}} = e^{- k \bar{α} d} [\begin{matrix} cos (k Δ α^{l} d / 2) & j sinh (k Δ α^{l} d / 2) \\ - j sinh (k Δ α^{l} d / 2) & cosh (k Δ α^{l} d / 2) \end{matrix}] .

In this regard, a circular diattenuator is given as

J_{CD} = e^{- k \bar{α} d} [\begin{matrix} cos (k Δ α^{c} d / 2) & j sinh (k Δ α^{c} d / 2) \\ - j sinh (k Δ α^{c} d / 2) & cosh (k Δ α^{c} d / 2) \end{matrix}] .

where Δα^c = α_l − α_r is the circular diattenuation coefficient; α_l and α_r are the attenuation coefficients for left and right circularly polarized light, respectively. On the other hand, recall that any Jones matrix J representing a nondepolarizing medium can be converted to a Mueller matrix M through the transformation

M = A (J \otimes J^{*}) A^{- 1},

where ⊗ indicates a direct product, and

A = [\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & - 1 \\ 0 & 1 & 1 & 0 \\ 0 & j & - j & 0 \end{matrix}] .

Let us disregard the constants multiplying Jones matrices in Eqs. (A.1–A.3). Mueller matrix for horizontal/vertical oriented diattenuator is, after applying transformation in Eq. (A.4),

M_{LD} = cosh (k Δ α^{l} d) [\begin{matrix} 1 & - tanh (k Δ α^{l} d) & 0 & 0 \\ - tanh (k Δ α^{l} d) & 1 & 0 & 0 \\ 0 & 0 & sech (k Δ α^{l} d) & 0 \\ 0 & 0 & 0 & sech (k Δ α^{l} d) \end{matrix}] .

Besides, Mueller matrix for linear diattenuator oriented at ±π/4 is, in this case,

M_{{LD}^{'}} = cosh (k Δ α^{l} d) [\begin{matrix} 1 & 0 & - tanh (k Δ α^{l} d) & 0 \\ 0 & sech (k Δ α^{l} d) & 0 & 0 \\ - tanh (k Δ α^{l} d) & 0 & 1 & 0 \\ 0 & 0 & 0 & sech (k Δ α^{l} d) \end{matrix}] .

Eqs. (A.6) and (A.7) are related through a rotation transformation. Finally, Mueller matrix for circular diattenuator is given as

M_{CD} = cosh (k Δ α^{c} d) [\begin{matrix} 1 & 0 & 0 & - tanh (k Δ α^{c} d) \\ 0 & sech (k Δ α^{c} d) & 0 & 0 \\ 0 & 0 & sech (k Δ α^{c} d) & 0 \\ - tanh (k Δ α^{c} d) & 0 & 0 & 1 \end{matrix}] .

The first row in matrices given by Eqs. (A.6–A.8) are related to the diattenuation vectors defined in Eqs. (13) and (26).

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Mueller imaging polarimetry of holographic polarization gratings inscribed in azopolymer films

Abstract

1. Introduction

2. Theory

2.1. Case I: ψ₁ = 0, ψ₂ = π/2 and ϕ = 0

2.2. Case II: ψ₁ = −ψ₂ = π/4 and ϕ = 0

2.3. Case III: ψ₁ = ψ₂ = ψ and ϕ = π/2

3. Materials and methods

4. Results

4.1. Case I: Horizontal and vertical linear polarizations

4.2. Case II: Linear polarizations oriented at ±π/4

4.3. Case III: Right and left circular polarization

5. Conclusion

Appendix

References and links

Cited By

Figures (8)

Equations (53)

Optics Express

Abstract

1. Introduction

2. Theory

2.1. Case I: ψ1 = 0, ψ2 = π/2 and ϕ = 0

2.2. Case II: ψ1 = −ψ2 = π/4 and ϕ = 0

2.3. Case III: ψ1 = ψ2 = ψ and ϕ = π/2

3. Materials and methods

4. Results

4.1. Case I: Horizontal and vertical linear polarizations

4.2. Case II: Linear polarizations oriented at ±π/4

4.3. Case III: Right and left circular polarization

5. Conclusion

Appendix

References and links

Cited By

Figures (8)

Equations (53)

Optics Express

2.1. Case I: ψ₁ = 0, ψ₂ = π/2 and ϕ = 0

2.2. Case II: ψ₁ = −ψ₂ = π/4 and ϕ = 0

2.3. Case III: ψ₁ = ψ₂ = ψ and ϕ = π/2