## Application theory of scattering and coupled mode analysis for liquid crystal diffractive grating |

Optics Express, Vol. 18, Issue 15, pp. 15513-15522 (2010)

http://dx.doi.org/10.1364/OE.18.015513

Acrobat PDF (1252 KB)

### 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

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]

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

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

_{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

_{i}d

_{k}= δε

_{ik}[10], where

_{0}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**′,

_{i}= ε

_{0}E

_{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}∣ ≪ ε

_{0}for nematic LC), we find,

**D**′ = 0 one can attain the next inhomogeneous Helmholtz equation [10] for the electrical field and vector electric induction of diffracted light.

**E**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,

**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

**R**∣

^{2}and higher, for the far field, during differentiation. Assume that incident light is plane wave

**E**(

**r**) =

**ê**E

_{0}(

**r**)exp(j

**k**·

**r**), where ê is unit vector of polarization. Considering that vectors electric induction

**D**′ = ε

**E**′ in the observation point, obtain from Eq. (6):

**Φ**is defined as

**Φ**= ∫(δε

**ê**)E

_{0}(

**r**)exp(−j

**q**·

**r**)dV and components of the vector are

**q**=

**k**′−

**k**.

**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).

*ϕ*is small or equal to zero. The components of electrical field contributing to diffraction in far field, derived from Eq. (7), are

_{y}d

_{y}and d

_{x}d

_{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**+ 0

**ẑ**.

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

_{0}and E

_{m}respectively, as the total electrical field E(

**r**).

_{0}plus the small anisotropic term ε

_{a}d

_{x}d

_{y}(∣ε

_{a}∣ ≪ ε

_{0}), of the periodic spatial oscillations, i.e. ε = ε

_{0}+ δε(

**r**) and δε = ε

_{a}d

_{x}d

_{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 ∂

^{2}E

_{0}/∂r

^{2}≪k∂E

_{0}/∂r, we find

**k**

_{m}−

**k**

_{m±(21−1)}±(2l−1)K

**x̂**(where l=2, 3, 4,⋯), we are lead to the next system of partial differential equations for the coupled modes.

^{2}ε

_{0}/c

^{2}= k

^{2}

_{0}= k

^{2}

_{1}= k

^{2}

_{−1}= ....k

^{2}

_{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/k

_{0}, or a condition of phase-synchronism in the x direction. Thus

_{0}≪ 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

_{0}(cosθ

_{m}−cosθ

_{m+1})z and k

_{0}(cosθ

_{m}−cosθ

_{m−1})z can be expended in a power series at small values K/k

_{0}. If m(K

^{2}/k

_{0})b ≪ 1 (the Raman-Nath regime) accumulated phase mismatch in z direction is negligible. Consequently exponential terms in Eq. (18) can be dismissed.

_{a}/(λε

^{1/2}

_{o}) and d = π/b

_{m}=1/2(J

_{m−1}− J

_{m+1}). Thus we can express electrical field in (19) through the Bessel function E

_{m}E = E

_{0}∣

_{z=0}(j)

^{m}J

_{m}(u) at u(z) = (2α/d)[1−cos(dz)]. Where electrical field is E

_{0}∣

_{z=0}= E(0). The electrical field of incident light will be

## 4. Theory of scattering and coupled mode analysis approximation

_{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:

_{y}= 0. We normalize the amplitude of incident light entering into the LC grating within the aperture such that, (E

_{0}(x,y,0)=1), and assume it is equal zero outside the aperture. Then integral Eq. (21) transforms to

**q**on the x and z axes are defined as q

_{x}=(2πn

_{0}/λ)sin(θ) and q

_{z}=(2π/λ)(n

_{e}−n

_{o})=(2π/λ)Δn.

_{0}= 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.

^{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]

## 5. Conclusion

## References and links

1. | Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. |

2. | M. Fritsch, H. Wohler, G. Haas, and D. Mlynski, “Liquid-Crystal Phase Modulator for Large Screen Projection,” IEEE Trans. Electron. Dev. |

3. | P. Shannon, W. Gibbons, S. Sun, and B. Swetlin, “Surface-Mediated Alignment of Nematic Liquid Crystals with Polarized Laser Light,” Nature |

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 |

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. |

8. | P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. |

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 , |

10. | L. D. Landau and E. M. Lifschitz, |

11. | A. Sommerfeld, |

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. |

**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

- Y. Hori, K. Asai, and M. Fukai, “Field Controllable Liquid-Crystal Light Valves,” IEEE Trans. Electron. Dev. 26(11), 1734–1737 (1979). [CrossRef]
- 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]
- 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).
- 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).
- W. E. Glenn, “The Use of Optical Diffraction to Produce Images,” J. SID 5(3), (1997).
- 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]
- G. Kreymerman, “Liquid crystal diffractive phase grating as light modulator for projection display,” Opt. Eng. 45(11), 116202 (2006). [CrossRef]
- P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982). [CrossRef]
- 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]
- L. D. Landau, and E. M. Lifschitz, Electrodynamics of Continuous Media (Oxford, England: Pergamon Press, 1984).
- A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, New York, 1949).
- 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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