1. Introduction

Early LC diffractive spatial light modulators (SLM) without polarizers [1, 2] were constructed by the patterned alignment of electrodes on a substrate with a uniform orientation of molecules of LC inside the cell. This structure has some disadvantages such as low diffraction efficiency, complexity in the alignment of electrodes with high density, and parasitic diffraction from patterned electrodes. Shanon et al. [3] proposed a patterned alignment of LC phase diffractive grating with an orthogonal orientation of LC molecules in alternating strips diffraction grating as a device independent of the polarization of the incident light. A drivable LC phase diffractive grating with reverse twisted LC molecules in alternating strips was introduced in [4] that is a device with high diffraction efficiency (close to 100%) as well as being independent of the polarization of the incident light. The SLM with reverse twisted LC grating embedded in the Schlieren system has been considered an alternative to the conventional LC light valve based on polarization modulation in a twisted nematic (TN)-LC cell placed between two polarizers. The structure of the reverse twisted diffraction grating is created by layers of LC with different orientations of LC molecules in alternating strips, as illustrated in Fig 1. The analysis of the data in [5–7] indicates that the LC diffractive light valve with reverse TN-LC grating, as a spatial modulator, allows the achievement of modulation efficiency close to 100%.

The common approach to determine the parameters of the diffracted electrical field for a LC grating is to use the matrix formalism of Jones calculus. This method is applied to one-dimensional structures and takes into account the twisted angle and birefringence. The Jones calculus provides only the phase and polarization state of the wave at each grid point on exit from the LC cell. Some numerical simulations based on Jones’ formalism have been developed for two-dimensional structures [8–9]. One common drawback of these methods is the computational demand. The Jones formalism does not take into consideration the deviation of the light beam from the direction of incident light due to diffraction inside the LC grating. The parameters of diffracted light in the far field are impossible to calculate using just the Jones formalism. This requires an additional approach that includes the methods of diffraction theory in the far field.

This work explores a method based on vector theory of scattering for a far field in combination with the coupled mode analysis within the LC grating. The method applied in this work is shown to be in satisfactory agreement with experiments.

2. Scattering theory approximation for nematic LC phase-diffraction grating

The tensor ε_ik is the tensor permittivity for the nematic liquid crystal and is defined as the sum of isotropic permittivity ε_o and an anisotropic term ε_a d_id_k = δε_ik [10], where

The values ε₀ and ε_a are two independent constants, δ_ik is the Kronecker delta function, d_k is the component of the vector director for the nematic crystal, and ε_a = ε_e − ε_o is the dielectric anisotropy of LC. The total field is the electrical field E of the incident wave and the electrical field E′ of the diffracted wave. We can express the electric induction of the incident light D and the diffracted light D′,

By substituting Eq. (1) into Eq. (2), considering that D_i = ε₀E_i, and neglecting for nematic LC the small terms δε_ik E′_k in Eq. (2), (i.e. the anisotropy of the permittivity is relatively small ∣ε_a∣ ≪ ε₀ for nematic LC), we find,

From Maxwell’s equations $\nabla \times \mathbf{E}\prime =\frac{j\omega }{c}\mathbf{H}\prime$ , $\nabla \times \mathbf{H}\prime =\frac{j\omega }{c}\mathbf{D}\prime$ and ∇ · D′ = 0 one can attain the next inhomogeneous Helmholtz equation [10] for the electrical field and vector electric induction of diffracted light.

The term δεE denotes the vector with components δε_ik E_k, and the wave number of the diffracted light is k′=(ω/c)ε^1/2 ₀. The Sommerfeld radiation boundary condition [11] at infinity is required in order to solve this equation uniquely. With this condition the solution of this equation for fare field is the convolution,

Where V is volume of the diffractive medium and G(ξ) = exp(jk′ξ)/4πξ is the Greens’s function, with ξ = ∣R−r∣ and F(r) = ∇×∇×(δε E). The variable R is the radius vector from some point inside the diffractive medium to the far field where the diffracted field is observed. The variable r is radius vector inside the diffracted region. Notice that wave vector k′ is parallel to R. Assume ∣r∣ ≪ ∣R∣ and ∣R−r∣ ≈ ∣R∣ − k̂′·r. Then equation (5) can be expressed as

We can dismiss terms of order 1/∣R∣² and higher, for the far field, during differentiation. Assume that incident light is plane wave E(r) = êE₀(r)exp(jk · r), where ê is unit vector of polarization. Considering that vectors electric induction D′ = εE′ in the observation point, obtain from Eq. (6):

Vector Φ is defined as Φ = ∫(δεê)E₀(r)exp(−jq · r)dV and components of the vector are

Where vector q = k′−k.

In order to eliminate R in future calculation, define the extinction coefficient per unit length or scattering coefficient h as the ratio of the total intensity of the light diffracted in all direction per unit volume of the diffracting medium to the incident flux density.

Were term Ω is solid angle. Placing Eq. (7) in Eq. (9) we obtain

Where ${\sin }^2\left(\psi \right)=\frac{{\mid \mathbf{k}\prime \times \left(\mathbf{k}\prime \times \hat{\mathbf{e}}\right)\mid }^2}{k^4}$

The angle ψ is the angle between vectors ê direction polarization of incident light and wave vector k′ of diffracted light. We can apply this approximation to the reverse TN-LC diffraction grating (Fig. 1) which has high diffraction efficiency. The diffraction grating is constructed by layers of LC with different orientations of LC molecules in alternating strips. This pattern has been achieved by using a rubbing technique and applying mask for LC layers to create a certain direction of molecules in each alternating strip. The vector director in adjacent strips of each LC layer has angles of same magnitude but in clockwise and counterclockwise directions, relative to y-direction. The angle of vector director gradually changes along z-direction (thickness of LC grating).

 

Fig. 1. The element (or pixel) of a reverse twisted LC phase diffractive grating with pre-tilt angle ϕ and zero voltage applied to the cell.

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We assume that the direction of incident light is normal to the surface of the LC grating and the pre-tilt angle ϕ is small or equal to zero. The components of electrical field contributing to diffraction in far field, derived from Eq. (7), are

The asynchronous terms with d_yd_y and d_xd_x do not contribute to diffraction and therefore were neglected in derivation of Eq. (11) and Eq. (12). In addition, the z component of vector Φ was ignored due to its small angle of diffraction. One can notice, for a reverse twisted LC grating, from Eq. (11) and Eq. (12) that linearly polarized the incident light along x axis detracted to the light with polarization along y axis and in contrary the incident light with polarization along y axis diffracted to light with polarization along x axis. If pre-tilt angle ϕ is zero the components of the vector director are d_x = cos(φ)Σ(x) and d_y = sin(φ). The angle φ is the twisting angle. We define the twisted angle as a nearest positive angle of vector director relative to x-axis. The mathematical expression for spatial profile of the vector director at zero tilt angle is:

d(r) = cos[φ(z)]Σ(x)x̂ + sin[φ(z)]ŷ + 0ẑ.

Assume that the twisting angle changes linearly with a thickness b of the LC layer, such that φ = πz/2b. The function Σ(x) can be presented as the square wave periodic function of variable x.

Where parameter K = 2π/L is the wave number of the diffraction grating and value L denotes the period of the grating.

It is necessary to define the amplitude of the electrical field for the incident light as a function of the coordinates inside the diffracted media (see Eq. (11) and Eq. (12)) in order to define diffracted electrical field in point of observation. Therefore the coupled mode analysis for multiple orders of diffraction is essential.

3. Coupled mode analysis for a reverse twisted nematic LC diffraction grating

The coefficients in equations (11) and (12) are exactly the same for two orthogonal polarizations of the incident light. Therefore the intensity of diffracted light is independent from the direction of polarization of incident light. As a result intensity distribution in diffraction maximums are the same for two orthogonal polarizations of incident light for reverse twisted LC grating. For the reverse TN-LC grating (Fig. 1) we can introduce the sum of the incident and diffracted plane waves, with amplitude E₀ and E_m respectively, as the total electrical field E(r).

Where m = 0, ±1, ±2, ±....., are the orders of the diffractive maximus and C.C. is the comlex conjugated terms. The independence intensity of diffracted light from polarization of the incident light allowed us to apply the next scalar paraxial wave equation for inhomogeneous medium, in order to define the electrical field of the incident light as function of coordinates in the LC grating.

The reverse TN-LC diffraction grating is a phase grating with dielectric permittivity which can be describe in scalar presentation by the constant term ε₀ plus the small anisotropic term ε_ad_xd_y(∣ε_a∣ ≪ ε₀), of the periodic spatial oscillations, i.e. ε = ε₀ + δε(r) and δε = ε_ad_xd_y =(1/2)ε_a sin(πz/b)Σ(x). Substituting Eq. (13) and Eq. (14) into Eq. (15) and neglecting the second derivative terms, at ∂²E₀/∂r² ≪k∂E₀/∂r, we find

If we assume that the incident light propagates along the z axis, and we keep only terms with closest phase matched exponential terms, ignoring definitely phase mismatched exponential terms of power k _m−k _m±(21−1)±(2l−1)Kx̂ (where l=2, 3, 4,⋯), we are lead to the next system of partial differential equations for the coupled modes.

Here we consider that ω²ε₀/c² = k² ₀ = k² ₁ = k² ₋₁ = ....k² _m in Eq. (16). The term K = 2π/L is the wave number of the diffraction grating along the x axis. The directions of the diffracted light are governed by the grating equation, sin(θ_m) = mK/k₀, or a condition of phase-synchronism in the x direction. Thus $K_x+(k_{x_m}-k_{x_{m+1}})=0$ and ${-K}_x+(k_{x_m}-k_{x_{m-1}})=0$ . The second term on the left side of equation (17) can be ignored for K/k₀ ≪ 1 (or angle diffraction θ_m -small) and at the aperture of an incident beam much greater than the thickness of the grating. The equations (17) will transform to

The phase-asynchronous terms k₀(cosθ_m−cosθ_m+1)z and k₀(cosθ_m−cosθ_m−1)z can be expended in a power series at small values K/k₀. If m(K²/k₀)b ≪ 1 (the Raman-Nath regime) accumulated phase mismatch in z direction is negligible. Consequently exponential terms in Eq. (18) can be dismissed.

Where α = ε_a/(λε^1/2 _o) and d = π/b

The system of differential equations (19) is similar to the recursion relation for the derivative of the Bessel functions J′_m=1/2(J_m−1 − J_m+1). Thus we can express electrical field in (19) through the Bessel function E_m E = E₀∣_z=0(j)^m J_m(u) at u(z) = (2α/d)[1−cos(dz)]. Where electrical field is E₀∣_z=0 = E(0). The electrical field of incident light will be

The thickness of the LC diffraction grating b is not an arbitrary parameter and it is defined by the boundary condition related to maximum diffraction efficiency at b = b₀ on exit from LC grating. This means u(b₀)=4αb₀/π=2.4048 or b₀ = 2.408π/4α for the case when the incident light on exit from the LC grating diminish to zero or will be 100% diffraction efficiency. The outcome of this boundary condition is E₀(b₀)=E(0)J₀(2.4048)=0.

4. Theory of scattering and coupled mode analysis approximation

When we substitute d_x = cos(φ)Σ(x), d_y = sin(φ) and Eq. (20) into integral of equations (11) or Eq. (12) we obtain the following expression:

Notice that diffraction takes place in the (x-z)-plane and the direction of the incident light along the z-axis and as a result component q_y = 0. We normalize the amplitude of incident light entering into the LC grating within the aperture such that, (E₀(x,y,0)=1), and assume it is equal zero outside the aperture. Then integral Eq. (21) transforms to

Where P(m) = (2m−1)(2π/L) and parameter “a” is the radius of the aperture. The projections of vector q on the x and z axes are defined as q_x =(2πn₀/λ)sin(θ) and q_z =(2π/λ)(n_e−n_o)=(2π/λ)Δn.

The parameter n₀ = ε^1/2 ₀ is the index refraction. The peaks of intensity of the diffracted light as function of angle diffraction θ is calculated from Eq. (23) and the experimental data for reverse twisted LC grating are presented in Fig. 2. We normalized both our calculation and experimental data to unity at maximum peak of intensity.

 

Fig. 2. The angular distribution of the diffraction maximums for a reverse TN-LC (E-7) diffractive grating; solid line (aperture of beam 0.5mm) and doted line (aperture of beam 0.2mm). The stems with boxes on top are experimental result. The grating parameters are; n_e = 1.7765, n_o=1.5225, b=2µm, L=75µm, pre-tilt angle 5°, wavelength of light 543nm.

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We see from the analytical analysis the existence only odd orders in the diffraction maximums, which are in satisfactory agreement with experiment. The occurrence of relatively small even orders of diffraction maximums in the experiment can be related to imperfections of the LC grating. For example, that can arise from deviations in the profile of the index refraction from a square wave function along x direction as well as non-optimized thickness of the LC grating. The domination of odd orders in the diffraction maximums as shown in Fig. 2 and the experimental data of Fig. 3 indicate that we have the opportunity to construct a diffractive SLM with close to 100% efficiency. Figure 2 illustrates a dependence on angular selectivity of the LC grating from the aperture of the incident light. The angular selectivity enhances with an increase aperture of light (reference peaks a solid line and doted line in Fig. 2). The application of coupled mode analysis for phase LC grating on its own does not allow calculations of dependency on angular selectivity and distribution intensity from aperture of light. In the presented model as well as in the general theory for a diffraction grating, the number of minimums between the principal maximums is equal to the number of periods of the grating, cover by aperture of light, minus one.

 

Fig. 3. The diffraction efficiency as function of alternative voltage (1 kHz) applied to electrodes of reverse TN-LC diffraction grating for two orthogonal polarization of incident light. The circles are for polarization parallel to vector director of LC molecules, while the diamonds are for polarization orthogonal to vector director. L=75µm, b=2µm, pre-tilt angle 5°.

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If we introduce a parameter of optical retardation, γ=2b(Δn/λ), which is the equivalent to that used for the design of a TN LC display, we can show in Fig. 4 the relation between diffraction transmission N∣B(γ)∣² and the optical retardation γ (N- coefficient normalized to unity). Here term B is from Eq. (22). We see a correlation between the curves for the diffraction transmission of the reverse TN-LC grating and the transmission of a typical 90° TN-LC cell placed between two conductive glass substrate with parallel-polarized plates on outside of LC cell. The transmission of a 90° TN-LC cell as function of retardation is governed by Gooch-Tarry formula [12] $T\left(\gamma \right)={\sin }^2\frac{\left[\frac{\pi }{2}{(1+{\gamma }^2)}^{\frac{1}{2}}\right]}{(1+{\gamma }^2)}$ , which is commonly used for design TN-LC display.

 

Fig. 4. The diffraction transmission N∣B(γ)∣² of a reverse TN LC (E-7) grating (solid line) and the transmission T(γ), of a 90°TN cell (dashed line) placed between parallel polarizers at zero voltage between electrodes as functions of optical retardation γ.

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We see (Fig. 4) that the position of the first minimum transmission (normally called black mode) for the Gooch-Tarry curve is matched to the maximum diffraction transmission of the reverse twisted LC grating (for the same LC material). The outcome of this correlation is that the thickness of the reverse twisted LC grating with maximum diffraction efficiency will be the same as for a conventional TN-LC cell at normal black mode with same LC media.

The pre-tilt angle was neglected in the previous consideration. This approximation allowed the diffraction efficiency to be independent from the direction polarization of incident light at the entrance to the reversed TN-LC grating (see Eq. (11) and Eq. (12)). In reality the pre-tilt angle is an essential parameter of LC devises, which is result of anchoring the ends of LC molecules (with designed uniform alignment) to the processed polyimide coated surface. The pre-tilt angle prevents reverse-tilted disclinations in the twisted and supertwisted nematic devices. However, an increase of the pre-tilt angle in reverse TN-LC grating make the diffraction efficiency dependent on the polarization of the incident light. The diffraction efficiency as function of pre-tilt angle measured in the experiment is shown in Fig. 5.

 

Fig. 5. The diffraction efficiency for the cells of a reverse TN-LC (E-7) grating for different pre-tilt angles 5° (L=75µm), 10° (L=50µm), 25° (L=24µm), 39° (L=50µm). The circles are for incident light with polarization parallel to vector director, the diamonds are for polarization orthogonal to vector director of LC molecules on entry to cells (b=2µm for all cells).

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The diffraction efficiency for incident light with polarization perpendicular to the vector director of LC molecules, on input to the diffraction grating, not change significantly with the raise of the pre-tilt angle. This phenomenon emerges because the index of refraction for this orientation of polarization does not change with increasing pre-tilt angle due to the orientation polarization along ordinary axis of LC. However, for incident light with polarization parallel to the vector director of LC molecules the diffraction efficiency is reduced significantly with increase pre-tilt angle over 10 degrees. The LC structure, with a pre-tilt angle, exhibits the double-refraction effect (at the entrance and exit of the LC cell) for two orthogonal polarized beams, which propagate along different optical paths. This is similar to the same effect in a birefringent material with optical axis oriented at a determined angle relative to the entrance surface. As a result the incident light with a direction of polarization in the plane of the pre-tilt vector director of LC and direction propagation of incident light will propagate a longer optical pass in LC than the incident beam with orthogonal polarization. The deviation in optical pass from optimal thickness is a cause for the reduced diffractive efficiency. This effect amplifies with an increase of pre-tilt angle. This is especially true when the incident angle, which is in plane formed by direction of grating’s stripes and direction of propagation of incident light, deviates from the normal, as shown in Fig. 6.

 

Fig. 6. The diffraction efficiency of reverse TN-LC (E-7) grating as function of incident angle for two cells with pre-tilt angles 5° - solid lines (L=75µm, b=2µm) and 10° - doted lines (L=50µm, b=2µm). The circles are for incident light with polarization parallel to vector director; the diamonds are for polarization orthogonal to vector director of LC molecules on entry to cell.

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Figure 3 illustrates insignificant change, for normal incident angle, in diffraction efficiency until a threshold voltage, about 2 volt, is reached. However, the threshold potential applied to the electrodes makes a LC cell with reverse grating more susceptible to incident angle, as illustrated in Fig. 7.

 

Fig. 7. The diffraction efficiency of a reverse TN-LC (E-7) grating as function of incident angle for the cell with a pre-tilt angle 5 degree at 2 volts AC (1 kHz) applied to electrodes of cell. The circles are for incident light with polarization parallel to vector director; the diamonds are for polarization orthogonal to vector director of LC molecules on entry to LC grating (L=75µm, b=2µm).

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The threshold electrical field applied to the TN-LC cell optimizes the optical path (maximum diffraction efficiency) at the normal incident angle. Nonetheless, the LC media exhibits the double refraction effect at the threshold voltage due to the tilting of the LC molecules. This effect increases the path for the incident light, which diverge from normal direction, with increasing tilt angle of the LC molecules. The outcome is a drastic reduction of diffraction efficiency, for two orthogonal polarized beams, when the path of the incident light deviates from the normal direction.

5. Conclusion

The detailed analysis of LC diffractive phase grating in this work allows one to optimize LC phase diffraction gratings in order to achieve maximum modulation efficiency. The occurrence of the dominant odd-diffraction orders for reverse twisted LC diffraction gratings tuned at optimum parameters indicates that the opportunity exists for the construction of an SLM with close to 100% efficiency. The analysis of the diffraction transmission as function of optical retardation (cell thickness) for reverse TN-LC diffraction grating correlate with Gooch-Tarry theory for the transmission of the conventional TN-LC cell placed between two polarizers. The experiment and theoretical analysis have shown that an SLM with reverse twisted LC diffraction grating is independent of the polarization of incident light for small pre-tilt angles. This approach combining theory of scattering and coupled mode analysis can be applied to a wider range of applications that includes phase matching Brag diffraction where angular selectivity depends from aperture of the incident light.