Abstract

The development and modeling of a liquid-crystal phase grating for real-time diffractive three-dimensional displays are discussed. The system being developed, which is called the ICVision system, utilizes a number of ideas that will result in a rugged, low-power three-dimensional display offering both vertical and horizontal parallax and eventually full color. Fringing fields created between interdigitated electrodes formed on top of VLSI die will induce a diffraction pattern in a thin layer of liquid crystal that will cover the die. A detailed electrostatic and diffraction analysis of liquid-crystal phase-grating regions that will make up the final display is given here. The electrostatic analysis is developed by use of the method of moments. The diffraction analysis is developed by use of rigorous coupled-wave diffraction theory. The numerical results obtrained from the mathematical model are compared with experimental diffraction results from preliminary LCD cells that have been assembled as prototype ICVision devices.

© 1995 Optical Society of America

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References

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  1. J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
  2. J. Prost, P. S. Pershan, “Flexoelectricity in nematic and smectic-A liquid crystals,” J. Appl. Phys. 47, 2298–2312 (1976).
    [CrossRef]
  3. G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
    [CrossRef]
  4. S. Chandrasekar, Liquid Crystals (Cambridge U. Press, England, 1992).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction- E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  6. K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
    [CrossRef]
  7. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
    [CrossRef]
  8. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimension coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
    [CrossRef]
  9. J. P. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Trans. Microwave Theory Tech. MTT-19, 547–555 (1971).
    [CrossRef]
  10. P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
    [CrossRef]
  11. D. Quak, G. den Boon, “Electric input-admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure,” IEEE Trans. Sonics Ultrason. SU-25, 44–50 (1978).
    [CrossRef]
  12. E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
    [CrossRef]

1991 (1)

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

1987 (3)

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimension coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
[CrossRef]

1985 (1)

P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
[CrossRef]

1983 (1)

1978 (1)

D. Quak, G. den Boon, “Electric input-admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure,” IEEE Trans. Sonics Ultrason. SU-25, 44–50 (1978).
[CrossRef]

1976 (1)

J. Prost, P. S. Pershan, “Flexoelectricity in nematic and smectic-A liquid crystals,” J. Appl. Phys. 47, 2298–2312 (1976).
[CrossRef]

1971 (1)

J. P. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Trans. Microwave Theory Tech. MTT-19, 547–555 (1971).
[CrossRef]

Chandrasekar, S.

S. Chandrasekar, Liquid Crystals (Cambridge U. Press, England, 1992).
[CrossRef]

Ciliax, R.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).

den Boon, G.

D. Quak, G. den Boon, “Electric input-admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure,” IEEE Trans. Sonics Ultrason. SU-25, 44–50 (1978).
[CrossRef]

Fritsch, M. W.

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimension coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction- E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

Ghijsen, W. J.

P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
[CrossRef]

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimension coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
[CrossRef]

Haas, G.

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

Kowel, S. T.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).

Kulick, J.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).

Leslie, T.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).

Mlynski, D. A.

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

Moharam, M. G.

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction- E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

Montgomery, J. P.

J. P. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Trans. Microwave Theory Tech. MTT-19, 547–555 (1971).
[CrossRef]

Mori, S.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Pershan, P. S.

J. Prost, P. S. Pershan, “Flexoelectricity in nematic and smectic-A liquid crystals,” J. Appl. Phys. 47, 2298–2312 (1976).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
[CrossRef]

Prost, J.

J. Prost, P. S. Pershan, “Flexoelectricity in nematic and smectic-A liquid crystals,” J. Appl. Phys. 47, 2298–2312 (1976).
[CrossRef]

Quak, D.

D. Quak, G. den Boon, “Electric input-admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure,” IEEE Trans. Sonics Ultrason. SU-25, 44–50 (1978).
[CrossRef]

Rokushima, K.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Tominaga, K.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Van Den Berg, P. M.

P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
[CrossRef]

Venema, A.

P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
[CrossRef]

Wohler, H.

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

Yamakita, J.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

D. Quak, G. den Boon, “Electric input-admittance of an interdigital transducer in a layered, anisotropic, semiconducting structure,” IEEE Trans. Sonics Ultrason. SU-25, 44–50 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. M. Van Den Berg, W. J. Ghijsen, A. Venema, “The electric-field problem of an interdigital transducer in a multilayered structure,” IEEE Trans. Microwave Theory Tech. MTT-33, 121–128 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space-time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

J. P. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Trans. Microwave Theory Tech. MTT-19, 547–555 (1971).
[CrossRef]

J. Lightwave Technol. (1)

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electrooptic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

J. Appl. Phys. (1)

J. Prost, P. S. Pershan, “Flexoelectricity in nematic and smectic-A liquid crystals,” J. Appl. Phys. 47, 2298–2312 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimension coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Mol. Crys. Liq. Crys. (1)

G. Haas, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two dimensional nematic director structures in inhomogeneous electric field,” Mol. Crys. Liq. Crys. 198, 15–28 (1991).
[CrossRef]

Other (3)

S. Chandrasekar, Liquid Crystals (Cambridge U. Press, England, 1992).
[CrossRef]

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision: A VLSI-based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
[CrossRef]

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Figures (20)

Fig. 1
Fig. 1

Two-dimensional array of integrated circuit die, showing overhead illumination and viewing zones.

Fig. 2
Fig. 2

Diffraction grating formed in LC overlay by fringing fields computed by a buried VLSI processor.

Fig. 3
Fig. 3

(a) Geometry of a unit cell of the ITO–LC device. (b) Simplified electrostatic model. Distance 2a is the width of the unit cell; b is the distance between the ground planes; 2c is the width of the ITO electrodes; h1 and h2 are the y positions of the ITO electrodes; ɛ _ 1 , ɛ _ 2 , and ɛ _ 3 are the dielectric permittivities of the regions of the unit cell as shown.

Fig. 4
Fig. 4

Geometry of the unit cell symmetry cases: (a) odd symmetry case, (b) even symmetry case.

Fig. 5
Fig. 5

Perpendicular homeotropic alignment.

Fig. 6
Fig. 6

Geometry of an ITO electrode LC diffraction grating: (a) no mirror, (b) mirror present. The ITO electrodes are assumed to be located at y = −s. A multilayer diffraction model is used to analyze diffraction. The diffraction grating of (a) is operated in a transmission mode.

Fig. 7
Fig. 7

Schematic of the experimental setup used to make the diffraction calculations.

Fig. 8
Fig. 8

(a) Electrostatic potential V(x, y) in volts, (b) electric field Ex(x, y) in volts per micrometer, (c) electric field Ey(x, y) in volts per micrometer that resulted from calculations shown in text. Here Eth = 0.3125 (V/μm), and the assumed LC layer width is 4 μm. The area marked Liquid Crystal Region in the inset is plotted in Figs. 912.

Fig. 9
Fig. 9

Relative optical dielectric permittivity element ɛxx − ɛo when interdigitated voltage V1 (see Fig. 8 inset) is (a) 3 V, (b) 5.4 V, (c) 7.8 V. Here ɛxx − ɛo was calculated from Eqs. (49) and (50) in the text. As the voltage increases, a larger amount of dielectric modulation occurs; ɛe = 0.583, ɛo = 1, 0 ≤ ɛxx − ɛo ≤ 0.538, and the peak value of (ɛxx − ɛo) = 0.583. The y-axis position y = 0 corresponds to y = h1 − 4 μm in the Fig. 8 inset. The ITO electrode edge is located at y = 4 μm here.

Fig. 10
Fig. 10

Relative optical dielectric permittivity ɛxy: (a) 3 V, (b) 5.4 V, (c) 7.8 V. The same conditions as in Fig. 9 apply, with −0.2915 ≤ ɛxy ≤ +0.2195.

Fig. 11
Fig. 11

Relative optical dielectric permittivity ɛyy − ɛo: (a) 3 V, (b) 5.4 V, (c) 7.8 V. The conditions in Fig. 9 apply, with 0 ≤ ɛyy − ɛo ≤ 0.583.

Fig. 12
Fig. 12

Optical dielectric permittivity tensor elements: (a) ɛxx − ɛo, (b) ɛxy, (c) ɛyy − ɛo. A disclination whose width is 20% of the interdigitated electrode spacing Λe = 2a is included in the modeling of ɛ _ _ . The disclination is centered between the electrodes; it was modeled when ψ = 0° [see Fig. 4(b)] was set in the disclination region and when Eq. (49) was used to model ɛ _ _ .

Fig. 13
Fig. 13

Electric fields Ex and Ey: (a), (b), nonsymmetric; (c), (d), symmetric. Two different voltage configurations are used to excite the interdigitated electrodes. The nonsymmetric electric fields are the same plots as shown in Figs. 8(b) and 8(c).

Fig. 14
Fig. 14

ɛxy dielectric permittivity element: (a) full view, (b) side view, symmetric, (c) side view, nonsymmetric voltage excitation. Note that in (b) ɛxy does not repeat itself over the interval −8 μm < x < 8 μm, whereas in (c) ɛxy repeats itself over the interval −8 μm < x < 0 and 0 ≤ x ≤ 8 μm.

Fig. 15
Fig. 15

Diffraction efficiency (DE) of the +2 and −2 orders when (a) calculated by the numerical model (rigorous coupled-wave theory) and (b) measured from the experiment. In (a) no disclinations were assumed to be present, ɛ1 = ɛo = ɛ2 = ɛ3 = 1, ɛe = 1.583, M = Δɛ/0.5(ɛo + ɛe) = 0.45, and θ = θa = 30°. In (b) ɛ1 ≅ ɛo ≅ ɛ2 = 2.3305, ɛe = 3.3054, and M = 0.35. The G refers to the curves where light was incident from the glass side, and I refers to curves where the light was incident from the ITO side.

Fig. 16
Fig. 16

Comparison of diffraction efficiency (0° incidence) results from (a) the experiment, and from the rigorous coupled-wave theory (b) with disclinations and (c) without disclinations.

Fig. 17
Fig. 17

Diffraction efficiency results of rigorous coupled-wave theory: (a) disclinations absent, and disclinations present and equal to (b) 10% electrode spacing, (c) 20% electrode spacing. Except for the disclination widths, the same numerical model parameters used in Figs. 15 and 16 were used. Plots of (a)–(c) may be directly compared with the experimental measured results of Fig. 15(b). Diffraction efficiency curves with θa = 0°, 30° angle of incidence were plotted together to show their relative size.

Fig. 18
Fig. 18

Zero-order diffraction efficiency shown in (a) is calculated by the numerical model (no disclinations present) of the paper and is shown as determined by experiment. The angle of incidence was θa = 0°, 10°, 20°, and 30°. The diffraction efficiency is shown when the light is incident upon either the glass side or the ITO side (see Fig. 15 inset). Both the glass side and the ITO side DE’s were so close for both the numerical and the experimental data that the curves can barely be distinguished. In the numerical calculation the numerical results were identical for incidence upon the glass or ITO electrode sides. The numerical model (rigorous coupled-wave theory) results were the same as for Figs. 1517.

Fig. 19
Fig. 19

Diffraction efficiency is plotted versus order (−8, −7, …, 0, …, 7, 8) and is plotted versus the voltage V1 [0 ≤ V1≤ 5 V, V2 = 0; see Fig. 4(b)]. In these figures λ = 0.6328 μm, ∊0 = ∊1 = 1. The angle of incidence in air was θ = θa = 30°. In (a), no mirror backed the LC–ITO diffraction grating. In (b)–(d), a mirror was assumed to back the ITO electrode LC grating. No disclinations were assumed to be present. M = 0.45.

Fig. 20
Fig. 20

Diffraction efficiency is shown when the electrode width and spacing are one half that of all other figures (width 2 μm, spacing 2 μm), λ = 0.6328 μm. A transmission and mirror case is shown. The plots were made by rigorous coupled-wave theory. M = 0.45.

Equations (89)

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V ( x , y ) x | x = 0 = 0 ,             V ( x , y ) x = a = 0.
V 1 ( x , y ) = m = 0 E m sin [ k x m ( x - a ) ] Y 1 m ( y ) ,
Y 1 m ( y ) = sinh ( k x m y ) , m = 0 , 1 , 2 , .
V 2 ( x , y ) = m = 0 B m sin [ k x m ( x - a ) ] Y 2 m ( y ) ,
Y 2 m ( y ) = sinh [ k x m ( b - y ) ] .
V 3 ( x , y ) x = c = V 0 ,             V 3 ( x , y ) x = a = 0.
V 3 ( x , y ) = V 3 H ( x , y ) + V 3 F ( x ) ,
V 3 H ( x , y ) x = c = 0 ,             V 3 H ( x , y ) x = a = 0 ,
V 3 F ( x ) x = c = V 0 ,             V 3 F ( x ) x = a = 0.
V 3 F ( x ) = - V 0 a - c x + a a - c V 0 = γ x + y .
V 3 ( x , y ) = - V 0 a - c x + a a - c V 0 + l = 1 sin [ k l ( x - c ) ] × { F l     cosh [ k l ( y - h 1 ) ] + D l sinh [ k l ( y - h 1 ) ] } .
E i = - V i = - x V i ( x , y ) a ^ x - y V i ( x , y ) a ^ y ,             i = 1 , 2 , 3 , .
V ( x , y ) = { V 2 ( x , y ) , h 2 y b V 3 ( x , y ) , h 1 y h 2 V 1 ( x , y ) , 0 y h 1 ,
V 2 ( x , h 2 ) = { V 0 , 0 x c V 3 ( x , h 3 ) , c x a .
0 a d x sin [ k x m ( x - a ) ] V 2 ( x , h 2 ) = 0 a d x sin [ k x m ( x - a ) ] V 0 + c a d x sin [ k x m ( x - a ) ] × [ V 3 F ( x ) + V 3 H ( x , h 2 ) ] .
0 a d x sin [ k x m ( x - a ) ] V 2 ( x , h 2 ) = B m α m ,
α m = [ Y 2 m ( h 2 ) a 2 ] - 1 .
V m f = 0 a d x sin [ k x m ( x - a ) ] V 0 + c a d x sin [ k x m ( x - a ) ] × ( - V 0 a - c x + a a - c V 0 ) .
c a d x sin [ k x m ( x - a ) ] V 3 H ( x , h 2 ) = c a d x sin [ k x m ( x - a ) ] ( l = 1 sin [ k l ( x - c ) ] × { F l     cosh [ k l ( h 2 - h 1 ) ] + D l sinh [ k l ( h 2 - h 1 ) } ) = ( I = 1 P m l { F l cosh [ k l ( h 2 - h 1 ) ] + D l sinh [ k l ( h 2 - h 1 ) ] } ) = V 32 m ,
P m l = c a d x sin [ k x m ( x - a ) ] sin [ k l ( x - c ) ] .
B m = α m ( V m f + V 32 m ) .
E m = β m ( V m f + V 31 m ) ,
V 31 m = l = 1 P m l F l ,
ɛ _ 2 E y 2 y = h 2 = ɛ _ 3 E y 3 y = h 2 ,
ɛ _ 2 { - m = 0 B m sin [ k x m ( x - a ) ] Y 2 m ( h 2 ) } = ɛ _ 3 ( - l = 1 sin [ k l ( x - c ) ] × { k l F l sinh [ k l ( h 2 - h 1 ) ] + k l D l cosh [ k l ( h 2 - h 1 ) ] } ) ,
Y 2 m ( y ) = y Y 2 m ( y ) .
m = 0 B m P m i Y 2 m ( h 2 ) = q 2 l { F l sinh [ k l ( h 2 - h 1 ) ] + D 1 cosh [ k l ( h 2 - h 1 ) ] } ,
ɛ _ 1 E y 1 y = h 1 = ɛ _ 3 E y 3 y = h 1 ,
ɛ _ 1 { - m = 0 E m sin [ k x m ( x - a ) ] Y 1 m ( h 1 ) } = ɛ _ 3 ( - l = 1 sin [ k l ( x - c ) ] × { k l F l sinh [ k l ( h 1 - h 1 ) ] + k l D l cosh [ k l ( h 1 - h 1 ) ] } ) ,
Y 1 m ( y ) = y Y 1 m ( y ) .
m = 0 E m P m l Y 1 m ( h 1 ) = q 1 l D l ,
V l U = l = 1 K l , l UL F l + l = 1 K l , l UR D l , V l U = - m = 0 α m V m f P m l Y 2 m ( h 2 ) ,
V l L = l = 1 K l , l LL F l + l = 1 K l , l LR D l , V l L = - m = 0 β m V m f P m l Y 2 m ( h 2 ) ,
K l , l UL = cosh [ k l ( h 2 - h 1 ) ] M 2 l , l - q 2 l sinh [ k 1 ( h 2 - h 1 ) ] δ l , l ,
K l , l UR = sinh [ k l ( h 2 - h 1 ) ] M 2 l , l - q 2 l cosh [ k 1 ( h 2 - h 1 ) ] δ l , l ,
K l , l LL = M 1 l , l ,
K l , l LR = - q 1 l δ l , l ,
M 1 l , l = m = 0 β m P m l P m l Y 1 m ( h 1 ) ,
M 2 l , l = m = 0 α m P m l P m l Y 2 m ( h 2 ) ,
V ( x , y ) x | x = 0 = 0 ,             V ( x , y ) x | x = a = 0.
X i = cos ( m π a x ) = cos ( k x m x ) ,             i = 1 , 3.
α m = [ ( 1 + δ 0 m ) a 2 Y 2 m ( h 2 ) ] - 1 , δ 0 m = { 1 , m = 0 0 , m 0 ,
β m = [ ( 1 + δ 0 m ) a 2 Y 1 m ( h 1 ) ] - 1 ,
Y 1 m ( y ) = { y L , m = 0 sinh ( m π a y ) , m 0 ,
Y 2 m ( y ) = { ( b - y ) L , m = 0 sinh [ m π a ( b - y ) ] , m 0 ,
P m l = c a d x cos ( m π a x ) sin [ k l ( x - c ) ] ,
k l = ( 1 - 1 2 ) π ( a - c ) ,             l = 1 , 2 , 3 , , V m f = V o 0 a d x cos ( m π a x ) = V o a δ 0 m ,
v ET = { V E ( x , y ) / V 0 , 0 < x < a V E ( a - x , y ) / V 0 , a < x < 2 a ,
v OT = { V O ( x , y ) / V 0 , 0 < x < a - V O ( a - x , y ) / V 0 , a < x < 2 a ,
V T ( x , y ) = V e v ET ( x , y ) + V o v OT ( x , y ) ,
ɛ _ _ = [ ɛ o 0 0 0 ɛ e 0 0 0 ɛ o ] ,
ɛ _ _ = [ ɛ o + Δ ɛ sin 2 ψ Δ ɛ sin ψ cos ψ 0 Δ ɛ sin ψ cos ψ ɛ o + Δ ɛ cos 2 ψ 0 0 0 ɛ o ] .
E = i S i ( y ) exp ( - j ψ i ) ,
H = 1 η o i U i ( y ) exp ( - j ψ i ) ,
- y S x i = - j k o U z i ,
y U z i = j k o ρ ( ɛ x x i - ρ S x p + ɛ x y i - ρ S y p ) ,
j k x i U z i = j k o ρ ( ɛ y x i - ρ S x ρ + ɛ y y i - ρ S y ρ ) ,
y ˜ S _ x - j K _ _ x S _ y = - j U _ z ,
y ˜ U _ z = j ( ɛ _ _ x x S _ x + ɛ _ _ x y S _ y ) ,
j K _ _ x U _ z = j ( ɛ _ _ y x S _ x + ɛ _ _ y y S _ y ) ,
S _ y = ɛ _ _ y y - 1 ( K _ _ x U _ z - ɛ _ _ y x S _ x ) .
y ˜ S _ x = j ( K x ɛ _ _ y y - 1 ɛ _ _ y x ) S _ x + j ( I _ _ - K _ _ x ɛ _ _ y y - 1 K _ _ x ) U _ z = A _ _ 11 S _ x + A _ _ 12 U _ z ,
y ˜ U _ z = j ( ɛ _ _ x x - ɛ _ _ x y ɛ _ _ y y - 1 ɛ _ _ y x ) S _ x + j ( ɛ _ _ x y ɛ _ _ y y - 1 K _ _ x ) U _ z = A _ _ 21 S _ x + A _ _ 22 U _ z .
V _ = [ S _ x U _ z ]
A _ _ = [ A _ _ 11 A _ _ 12 A _ _ 21 A _ _ 22 ] ,
y ˜ V _ = A _ _ V _ .
E n e = i S _ i n ( y ˜ ) exp ( q n y ˜ - j ψ ˜ i ) ,
H n e = 1 η o i U _ i n ( y ˜ ) exp ( q n y ˜ - j ψ i ) .
E 2 = n C n E n e ,
H 2 = n C n H n e ,
H z 1 inc = E 0 η 0 exp [ - j ( k x 0 x - k y 0 y ) ] δ i 0 ,
E x 1 inc = E 0 n 1 ɛ 1 cos ( θ ) exp [ - j ( k x 0 x - k y 0 y ) ] δ i 0 ,
H z 1 ref = E 0 η 0 i r i exp [ - j ( k x i x + k y 1 i y ) ] ,
E x 1 ref = - E 0 ɛ 1 i r i k y 1 i exp [ - j ( k x i x + k y 1 i y ) ] ,
k y 1 i = { ( n 1 2 k 0 2 - k x i 2 ) 1 / 2 , n 1 k 0 > k x i - j ( k x i 2 - n 1 2 k 0 2 ) 1 / 2 , n 1 k 0 < k x i } = k ˜ y 1 i k o .
H z 3 tr = E 0 η 0 i t i exp { - j [ k x i x - k y 3 i ( y + s ) ] } ,
E x 3 tr = E 0 ɛ 3 i t i k y 3 i exp { - j [ k x i x - k y 3 i ( y + s ) ] } ,
k y 3 i = { ( n 3 2 k 0 2 - k x i 2 ) 1 / 2 , n 3 k 0 > k x i - j ( k x i 2 - n 3 2 k 0 2 ) 1 / 2 , n 3 k 0 < k x i } = k ˜ y 3 i k o .
( E x 1 inc + E x 1 ref ) y = 0 + = E x 2 y = 0 - ,
( H z 1 inc + H z 1 ref ) y = 0 + = H z 2 y = 0 - ,
E x 3 tr y = - s - = E x 2 y = - s + ,
H z 1 tr y = - s - = H z 2 y = - s + .
E 0 δ i 0 + r i = n C n U zin ,
1 ɛ 1 ( n 1 cos θ δ i 0 - k ˜ y 1 i r i ) = n C n U zin ,
t i = n C n U zin exp ( - q n s ˜ ) ,
1 ɛ 3 k ˜ y 3 i t i = n C n U zin exp ( - q n s ˜ ) ,
2 E 0 n 1 cos θ δ i 0 = n C n ( k ˜ y 1 i ɛ 1 U zin + S zin ) ,
0 = n C n ( - k ˜ y 3 i ɛ 3 U z i n + S z i n ) exp ( - q n s ˜ ) ,
η = P ORDER P INCIDENT × 100 % .

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