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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 15 — Jul. 19, 2010
  • pp: 15513–15522
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Application theory of scattering and coupled mode analysis for liquid crystal diffractive grating

Grigoriy Kreymerman  »View Author Affiliations


Optics Express, Vol. 18, Issue 15, pp. 15513-15522 (2010)
http://dx.doi.org/10.1364/OE.18.015513


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Abstract

This work presents a detailed analysis of a liquid crystal (LC) phase diffraction grating based on a approach combining vector theory of scattering and coupled mode analysis. In general, the coupled mode analysis gives a solution for the diffracted field regardless of aperture and the polarization state of the incident light. However, the aperture of the incident light defines the angular selectivity of the diffraction grating as well as the distribution of the intensity of the diffractive maximums. The solution of the vector theory of scattering in combination with the coupled mode analysis for diffraction of the light beam with finite aperture has allowed one to optimize the parameters of the high efficiency diffractive LC grating. The analytic solutions here were verified with experimental results for a reverse-twisted LC grating and a comparison with the standard Gooch-Tarry’s method, which typically applied for a twisted nematic LC display.

© 2010 Optical Society of America

1. Introduction

Early LC diffractive spatial light modulators (SLM) without polarizers [1

1. Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26 (11), 1734–1737 (1979). [CrossRef]

, 2

2. M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36 (9), 1882–1887 (1989). [CrossRef]

] 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

3. P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

] 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

4. P. J. Bos, J. Chen, J. W. Doane, B. Smith, C. Holton, and W. Glenn, “An Optically Active Diffractive Device for a High Efficiency Light Valve”, Digest of Technical Papers, Society for Information Display International Symposium, pp. 601–604. (1995)

] 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

5. W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5 (3), (1997).

] 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

8. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72 (4), 507–513 (1982). [CrossRef]

]. 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 εik is the tensor permittivity for the nematic liquid crystal and is defined as the sum of isotropic permittivity εo and an anisotropic term εa didk = δεik [10

10. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984)

], where

εik=ε0δik+εadidk
(1)

The values ε0 and εa are two independent constants, δik is the Kronecker delta function, dk is the component of the vector director for the nematic crystal, and εa = εe − εo 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′,

Di+Di=εik(Ek+Ek)
(2)

By substituting Eq. (1) into Eq. (2), considering that Di = ε0Ei, and neglecting for nematic LC the small terms δεik E′k in Eq. (2), (i.e. the anisotropy of the permittivity is relatively small ∣εa∣ ≪ ε0 for nematic LC), we find,

Di=ε0Ei+δεikEk
(3)

2D+k2D=××(δεE)
(4)

The term δεE denotes the vector with components δεik Ek, and the wave number of the diffracted light is k′=(ω/c)ε1/2 0. The Sommerfeld radiation boundary condition [11

11. A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949)

] 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,

D(R)=VG(Rr)F(r)dr
(5)

Where V is volume of the diffractive medium and G(ξ) = exp(jk′ξ)/4πξ is the Greens’s function, with ξ = ∣Rr∣ 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 ∣Rr∣ ≈ ∣R∣ − ′·r. Then equation (5) can be expressed as

D(R)=14π××(exp(jkRR)(δεE)exp(jk·r)dV
(6)

We can dismiss terms of order 1/∣R2 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):

E=k×(k×Φ)exp(jkR)4πRε
(7)

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

Φi=εadidkekE0(r)exp(jq·r)dV
(8)

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.

h=E2R2dΩVE(0)2
(9)

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

h=k4sin2(ψ)16π2ε2VE(0)2Φ2dΩ
(10)

Where sin2(ψ)=k×(k×ê)2k4

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.

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

Ey=(k)2exp(jkR)4πεRεadydxE0x(r)exp(jq·r)dV
(11)
Ex=(k)2exp(jkR)4πεRεadxdyE0y(r)exp(jq·r)dV
(12)

The asynchronous terms with dydy and dxdx 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 dx = cos(φ)Σ(x) and dy = 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) + sin[φ(z)]ŷ + 0.

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.

(x)=4π[n=1(1)n+12n1cos[(2n1)Kx]]
(13)

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 Em respectively, as the total electrical field E(r).

E(r)=12mEm(r)exp(jkm·r)+C.C.
(14)

ΔE+(εω2c2)E=0
(15)

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 εadxdy(∣εa∣ ≪ ε0), of the periodic spatial oscillations, i.e. ε = ε0 + δε(r) and δε = εadxdy =(1/2)εa 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

m[12(2jkm·Emkm2Em)exp(jkm·r)+C.C.]
+ω2c2[ε0+εasin(πzb)πn=1(1)n+12n1exp[j(2n1)Kx]+C.C.]
×m[12Emexp(jkm·r)+C.C.]=0
(16)

cosθmEmz+sinθmEmx=jεasin(πzb)λε012[Em+1exp(jKx)exp[j(kmkm+1)·r]+Em1exp(jKx)exp[j(kmkm1)·r]]
(17)

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(θm) = mK/k0, or a condition of phase-synchronism in the x direction. Thus Kx+(kxmkxm+1)=0 and Kx+(kxmkxm1)=0 . The second term on the left side of equation (17) can be ignored for K/k0 ≪ 1 (or angle diffraction θm -small) and at the aperture of an incident beam much greater than the thickness of the grating. The equations (17) will transform to

Emz=jεasin(πzb)λεo12[Em+1exp(jk0(cosθmcosθm+1)z)+Em1exp(jk0(cosθmcosθm1)z)]
(18)

The phase-asynchronous terms k0(cosθm−cosθm+1)z and k0(cosθm−cosθm−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.

Emz=jαsin(dz)(Em+1+Em1)
(19)

Where α = εa/(λε1/2 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(Jm−1 − Jm+1). Thus we can express electrical field in (19) through the Bessel function Em E = E0z=0(j)m Jm(u) at u(z) = (2α/d)[1−cos(dz)]. Where electrical field is E0z=0 = E(0). The electrical field of incident light will be

E0=E(0)J0(u)
(20)

4. Theory of scattering and coupled mode analysis approximation

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

E0(x,y,z)dydxexp(jq·r)dV
=E0(x,y,0)J0[u(z)][sin(πzb)2](x)exp[j(qxx+qzz)]dxdydz
(21)

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 qy = 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

aadyaa(x)exp(jqxx)dxA0bJ0(u)sin(πzb)exp(jqzz)dzB=2aAB
(22)
A=8Lm=1(1)m+1P2(m)qx2[2cos(qxa)sin[P(m)a]qxP(m)2sin(qxa)cos[P(m)a]]
(23)

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 qx =(2πn0/λ)sin(θ) and qz =(2π/λ)(ne−no)=(2π/λ)Δn.

The parameter n0 = ε1/2 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; ne = 1.7765, no=1.5225, b=2µm, L=75µm, pre-tilt angle 5°, wavelength of light 543nm.

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°.

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

12. C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8 (13), 1575–1584 (1975). [CrossRef]

] T(γ)=sin2[π2(1+γ2)12](1+γ2) , 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.

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.

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).

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.

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).

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.

References and links

1.

Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26 (11), 1734–1737 (1979). [CrossRef]

2.

M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36 (9), 1882–1887 (1989). [CrossRef]

3.

P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).

4.

P. J. Bos, J. Chen, J. W. Doane, B. Smith, C. Holton, and W. Glenn, “An Optically Active Diffractive Device for a High Efficiency Light Valve”, Digest of Technical Papers, Society for Information Display International Symposium, pp. 601–604. (1995)

5.

W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5 (3), (1997).

6.

C. E. Holton, P. Bos, M. Miller, and W. Glenn, “Patterned Alignment Liquid Crystal Diffractive Spatial Light Modulators and Devices,” SPIE Proc.3292–04, Photonics West, San Jose, CA, Jan., (1998) [CrossRef]

7.

G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45 (11), 116202 (2006). [CrossRef]

8.

P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72 (4), 507–513 (1982). [CrossRef]

9.

K. H. Yang and M. Lu, “Nematic LC modes and LC phase grating for reflective spatial light modulators,” IBM J. Res. Develop. High-resolution displays , 42, 401 (1998). [CrossRef]

10.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984)

11.

A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949)

12.

C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8 (13), 1575–1584 (1975). [CrossRef]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(230.3720) Optical devices : Liquid-crystal devices
(250.4110) Optoelectronics : Modulators

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 7, 2010
Revised Manuscript: June 16, 2010
Manuscript Accepted: June 17, 2010
Published: July 7, 2010

Citation
Grigoriy Kreymerman, "Application theory of scattering and coupled mode analysis for liquid crystal diffractive grating," Opt. Express 18, 15513-15522 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15513


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References

  1. Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979). [CrossRef]
  2. M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. 36(9), 1882–1887 (1989). [CrossRef]
  3. P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature 351, 351–352 (1991).
  4. P. J. Bos, J. Chen, J. W. Doane, B. Smith, C. Holton, and W. Glenn, “An Optically Active Diffractive Device for a High Efficiency Light Valve”, Digest of Technical Papers, Society for Information Display International Symposium, pp. 601–604 (1995).
  5. W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).
  6. C. E. Holton, P. Bos, M. Miller, and W. Glenn, “Patterned Alignment Liquid Crystal Diffractive Spatial Light Modulators and Devices,” SPIE Proc. 3292–04, Photonics West, San Jose, CA, Jan., (1998). [CrossRef]
  7. G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006). [CrossRef]
  8. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982). [CrossRef]
  9. K. H. Yang, and M. Lu, “Nematic LC modes and LC phase grating for reflective spatial light modulators,” IBM J. Res. Develop. High-resolution displays, 42, 401 (1998). [CrossRef]
  10. L. D. Landau, and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984).
  11. A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949).
  12. C. H. Gooch, and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975). [CrossRef]

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