lcd display module

REFERENCES 45 [Campbell F. W., 1968] F. W. Campbell and J. G. Robson, Application of Fourier analysis to the visibility of gratings, J. Physiol. (London), 197, pp.551–566, 1968. [Dewald D. S., 2001] D. Scott Dewald, S. M. Penn, and M. Davis, Sequential color recapture and dynamic filtering, SID’01, Digest, p.1076, 2001. [Doany F. E., 1998] F. E. Doany, R. N. Singh, A. E. Rosenbluth, and G. L.–T. Chiu, Projection display throughput efficiency of optical transmission and light-source collection, IBM J. Res. Dev, 42, pp.387–399, 1998. [Duelli M., 2002] M. Duelli, T. McGettigan, and C. Pentico, Integrator rod with polarization recycling functionality, SID’02, Digest, p.1078, 2002. [Fisher E., 1992] E. Fisher and H. Hoerster, High pressure mercury vapor discharge lamp, US Patent 5,109,181, 1992. [Fisher E., 1998] E. Fisher, Ultra high performance discharge lamps for projection TV systems, 8th International Symposium on Light Sources, Greifswald, Germany, 1998. 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MacAdam, Transformations of I.C.I. color specifications, J. Opt. Soc. Am., 17, p.294, 1937. [McGettigan T., 2004] T. McGettigan, Etendue and its application in light engine lumens budgeting, SID’04, Applications Tutorials, p.1, 2004. [Moench H., 2000] H. Moench, G. Derra, E. Fischer, and X. Riederer, ARC stabilisation for short arc projection lamps, SID’00, Digest, p.84, 2000. [Moskovich J., 1997] J. Moskovich, Telecentric lens systems for forming an image of an object composed of pixels, US Patent 5,625,495, 1997. [Moskovich J., 2003] J. Moskovich, Compact, telecentric projection lenses for use with pixelized panels, US Patent 6,563,650, 2003. [Nagata Y., 2004] Y. Nagata, A. Kagotani, and K. Ebina, An advanced projection screen with a wide vertical view angle, SID’04 Digest, p.846, 2004. [Peterson M., 2004] M. Peterson, D. Slobodin, J. Gohman, and S. Bierhuizen, Rear projection display system, US Patent 6,728,032, 2004. [Robson J. G., 1966] J. G. Robson, Spatial and temporal contrast sensitivity functions of the visual system, J. Opt. Soc. Am., 56, pp.1141–1142, 1966. [Shimizu J. A., 2001] J. A. Shimizu, Scrolling color LCOS for HDTV rear projection, SID’01, Digest, p.1072, 2001. [Shimizu Y., 2003] Y. Shimizu, S. Iwata, T. Yoshida, S. Takahashi, and T. Abe, A fine-pitch screen for rear projection TV, SID’03 Digest, p.886, 2003. [Stupp E. H., 1999] E. H. Stupp and M. S. Brennesholtz, Projection Display, John Wiley & Sons, Ltd, Chichester, 1999. [Umeya M., 2004] M. Umeya, M. Hatano, and N. Egashira, New front-projection screen comprised of cholesteric-LC films, SID’04, Digest, p.842, 2004. [Wang S., 1988] S. Wang and L. Ronchi, Principles and design of optical array, in E. Wolf (ed.), Progress in Optics, 25, pp.279–347, North-Holland, Amsterdam, 1988. 46 LC PROJECTION SYSTEM BASICS [Williamson S. J., 1983] S. J. Williamson and H. Z. Cummins, Light and color in nature and art, John Wiley & Sons, Inc., New York, 1983. [Wolfe C. R., 2004] C. R. Wolfe, M. Paukshto, and p.Smith, Design and performance of high contrast polarized light front-projection screens, SID’04, Digest, p.838, 2004. [Wright W. D., 1928] W. D. Wright, A re-determination of the trichromatic coefficients of the spectral colours, Trans. Opt. Soc., 30, pp.141–164, 1928. [Wyszecki G., 1982] G. Wyszecki and W. S. Stiles, Color Science, John Wiley & Sons, Inc., New York, 1982. 3 Polarization Basics 3.1 Introduction Polarization control is fundamental to LCD projection. The understanding of the basic properties and mathematical representations of polarization is therefore essential. This chapter introduces the various mathematical constructs that describe the polarization of light and presents the methods by which optical materials manipulate the state of polarization. It covers the basic electromagnetic theory of plane waves, polarization representations, and the relationship between them 7 inch lcd display module. Interaction with isotropic materials will be briefly introduced as a precursor to propagation through anisotropic materials. The concepts of birefringence and retardance will be reviewed. The last sections will be devoted to the mathematical modeling of polarization for light propagating through multiplayer anisotropic media. 3.2 Electromagnetic Wave Propagation Electromagnetic theory describes light as a propagating electromagnetic wave [Born M., 1980, chapter 1]. It generally requires four basic field vectors for its complete description: the electric field strength E, the electric displacement D, the magnetic field H, and the magnetic flux density B. Of these four vectors lcd monitor, E, which vibrates in time and space as the beam propagates, is chosen to define the state of polarization of light waves. Once the polarization of E has been determined, the three remaining field vectors D(cid:1) H, and B can be found through Maxwell’s field equations and associated material relations. In an isotropic medium, the direction of vibration is always orthogonal to the direction of propagation. For this transverse mode, there are two independent directions of vibration, which we can choose arbitrarily. Fourier analysis of the temporal (t) and spatial (r) variation of a light wave’s electric field E (cid:2)r(cid:1) t(cid:3) yields spectral components with frequencies from ultraviolet (UV) to Polarization Engineering for LCD Projection M. G. Robinson, J. Chen and G. D. Sharp © 2005 John Wiley & Sons, Ltd POLARIZATION BASICS 48 infrared (IR). Visible light extends from 4× 1014 to 8× 1014 hertz (Hz). For projection applications, we can assume that the system is linear relative to light intensity. This means that we can treat the transmission or reflectance of light independently for each wavelength. In addition, we can also calculate the optical response independently for each angle of incidence. Complete solutions then constitute a linear sum of the individual monochromatic plane wave solutions. In this manner, we need only consider in this book the mathematical manipulation of monochromatic plane optical waves. 3.2.1 Polarization of Monochromatic Waves [Born M., 1980; Yeh P., 1988; Azzam R. M. A., 1989; Yeh P., 1999] A monochromatic plane wave propagating in an anisotropic and homogeneous medium can be written: E = A cos(cid:2)(cid:4)t − k· r(cid:3) (3.1) where (cid:4) is the angular frequency, k is the wavevector, and A is a constant vector representing the amplitude. The magnitude of the wavevector k is related to the frequency by the equation: k = n(cid:4)/c = 2(cid:5)n/(cid:6) (3.2) where n is the refractive index of the medium, c is the speed of light in vacuum, and (cid:6) is the wavelength of light in vacuum. The refractive index in general depends upon wavelength, which is known as dispersion. It is a real number for transparent materials, but complex for materials that absorb. Since electromagnetic waves are transverse in nature, the electric field vector is always perpendicular to the propagation vector; that is: k· E = 0 (3.3) From Maxwell’s equations, it can be proven that the time evolution of the electric field vector for monochromatic light is sinusoidal; that is, the electric field oscillates with a single frequency. Assuming a light wave propagating along the z-axis, the electric field vector must lie in the xy plane. In this way, two mutually independent components of the electric field vector can be expressed as: Ex = Ax cos(cid:2)(cid:4)t− kz+ (cid:7)x(cid:3) Ey = Ay cos(cid:2)(cid:4)t− kz+ (cid:7)y(cid:3) (3.4) We have used two independent positive amplitudes Ax(cid:1) Ay and two independent phases (cid:7)x(cid:1) (cid:7)y to reflect the mutual independence of the two components. Since amplitude is positive by definition, the phase angles are defined within the range −(cid:5) < (cid:7) ≤ (cid:5). The locus of two independent orthogonal oscillations at the same frequency is an ellipse, which is analogous to the classic motion of a two-dimensional harmonic oscillator. After several steps of elementary algebra that eliminates (cid:2)(cid:4)t− kz(cid:3) from equation (3.4), we obtain: = sin2 (cid:7) (cid:2)2 − 2ExEy cos (cid:7) (cid:2)2 + (3.5) (cid:1) (cid:1) Ex Ax Ey Ay AxAy ELECTROMAGNETIC WAVE PROPAGATION 49 y′ y a φ b ε x′ x Figure 3.1 Polarization ellipse of monochromatic light (3.6) (3.7) where (cid:7) = (cid:2)(cid:7)y − (cid:7)x(cid:3) and −(cid:5) < (cid:7) ≤ (cid:5). By rotating coordinates, we are able to simplify equation (3.5). Let x(cid:2) be the new set of axes along the principal axes of the ellipse as shown in Figure 3.1. The equation of the ellipse in this new coordinate system becomes: and y(cid:2) (cid:1) (cid:2)2 + (cid:1) E(cid:2) x a E(cid:2) y b (cid:2)2 = 1 where: y sin2 (cid:8)+ 2AxAy cos (cid:7) cos (cid:8) sin (cid:8) y cos2 (cid:8)− 2AxAy cos (cid:7) cos (cid:8) sin (cid:8) x cos2 (cid:8)+ A2 a2 = A2 x sin2 (cid:8)+ A2 b2 = A2 tan(cid:2)2(cid:8)(cid:3) = 2AxAy x − A2 A2 y cos (cid:7) Complete specification of the elliptical polarization requires: 1. The azimuthal angle (cid:8) (−(cid:5)/2 to (cid:5)/2). The angle between the major axis of the ellipse and the x-axis. 2. The ellipticity e. The ratio of the length of the semi-minor axis of the ellipse b to the length of its semi-major axis a; that is: e = b a (3.8) 3. The handedness of the polarization ellipse. The polarization is right-handed if the ellipse rotates in a clockwise sense, and left-handed if the ellipse rotates in an anti-clockwise sense, when looking along the direction of propagation at any given position in space. It is convenient to incorporate the handedness in the definition of the ellipticity e by allowing the ellipticity to assume positive and negative values, corresponding to right-handed and left-handed polarizations, respectively. The ellipticity angle is defined as: (cid:9) = tan −1(cid:2)e(cid:3) (3.9) 50 POLARIZATION BASICS where (cid:9) is limited to ±(cid:5)/4. Since e relates directly to (cid:7), its sign also defines the handedness of the polarization state, such that (cid:7) > 0 corresponds to left-handed, while (cid:7) < 0 is right-handed. 4. The amplitude. A measure of the strength of vibration. Its square is proportional to the energy density of the wave at any fixed point of observation: A = (cid:2)a2 + b2(cid:3)1/2 (3.10) 5. The absolute phase. The angle between the initial position of the electric field vector at time t = 0 and the major axis of the ellipse. Since all detectors respond in a time average manner, only relative phase is of consequence to the problems faced in this book. Figure 3.2 shows polarization ellipses for various phase angles (cid:7)(cid:2)Ax = Ay(cid:3). As we can see, the circular and linear states of polarization (SOPs) are the special cases of the more general state of elliptical polarization, and are generated when the ellipticity, e, assumes the special values of ±1 and 0, respectively. It happens that circular and linear SOPs are the most important states in an LCD projection system. For example, in order to achieve good contrast and maximum brightness in a reflective LCOS projection system, the black SOP at the mirror must be linear while the white SOP must be circular. Since the LC is sandwiched between crossed polarizers in a transmissive HTPS projection system, the SOP should be linear and matched to the front polarizer to ensure a good contrast [Chen J., 2004]. δ = –7π/8 δ = –3π/4 δ = –5π/8 δ = –π/2 δ = –3π/8 δ = –π/4 δ = –π/8 δ = 0 δ = π/4 δ = π/8 Figure 3.2 Polarization ellipses at various phase angles under assumption of Ax = Ay in equation (3.4) δ = 3π/4 δ = 7π/8 δ = 3π/8 δ = 5π/8 δ = π/2 δ = π 3.2.2 Complex Number Representation [Yeh P., 1988; Azzam R. M. A., 1989; Yeh P., 1999] From equation (3.7), a general SOP, which is independent of overall intensity, is completely defined if we know the ratio of Ax to Ay, and relative phase (cid:7). A convenient two-dimensional representation of the SOP uses a single complex number (cid:10), which is defined as: (cid:10) = ei(cid:7) tan (cid:11) = Ay Ax ei(cid:2)(cid:7)y−(cid:7)x(cid:3) (3.11) ELECTROMAGNETIC WAVE PROPAGATION 51 Im(χ) Re(χ) Figure 3.3 Polarization states to different points in complex plane of polarization where the angle (cid:11) is confined to the range between 0 and (cid:5)/2. Once again, the SOP can be expressed entirely by (cid:7) and (cid:11) , or simply the complex number (cid:10). Figure 3.3 illustrates various polarization states in the complex plane. It can be seen that each point in the complex plane represents a unique polarization state. Left-handed elliptical polarization states are in the upper half of the plane, while right-handed ones are in the lower half. The states along the real axis are linearly polarized with various inclination angles. The relationship between the two SOP representations can be determined from simple mathematical manipulation. From equation (3.7), the inclination angle (cid:8) and the ellipticity angle (cid:9)(cid:2)(cid:9) = tan −1(cid:2)e(cid:3)(cid:3) are simply related to (cid:10) by: tan(cid:2)2(cid:8)(cid:3) = 2Re(cid:2)(cid:10)(cid:3) 1−(cid:3)(cid:10)(cid:3)2 sin(cid:2)2(cid:9)(cid:3) = − 2Im(cid:2)(cid:10)(cid:3) 1+(cid:3)(cid:10)(cid:3)2 = tan(cid:2)2(cid:11)(cid:3) cos (cid:7) = − sin(cid:2)2(cid:11)(cid:3) sin (cid:7) Orthogonal polarization states satisfy the relationships: (cid:10)∗(cid:10)orth = −1 (cid:10)orth = −(cid:10) (cid:3)(cid:10)(cid:3)2 = − ei(cid:7) tan (cid:11) (3.12) (3.13) (3.14) 3.2.3 Jones’ Vector Representation [Yeh P., 1988; Azzam R. M. A., 1989; Yeh P., 1999] Whilst the representations of the SOP so far presented are very concise, they do not contain information regarding intensity. To investigate the brightness and contrast of an LCD pro- jection system it is clearly necessary to calculate intensities. The Jones’ vector, introduced by R. C. Jones [Jones R. C., 1941], is a more convenient means of describing both the SOP and the intensity of a plane wave. Jones’ matrices that act to propagate Jones’ vectors are