Abstract

A mathematical model with experimental verification is presented to characterize the performance of surface and embedded electrodes in 2-D electrooptic modulators. From the solution of a discretized integral equation for the electrode surface charge, the electrode capacitance and the electric field penetration and uniformity are related to the switching voltage, speed, and uniformity of the electrooptic modulation. Fabricated surface and embedded electrodes in 9/65/35 PLZT are then evaluated with respect to the predictions of the model and the saturated quadratic response of the electrooptic material. These results provide important insight into the design trade-offs of switching speed, halfwave voltage, switching energy, and modulation uniformity of surface and embedded modulator geometries.

© 1990 Optical Society of America

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References

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  1. J. D. Harris, V. Luft, “Electrode Slotting Process for Thin PLZT Wafers,” Ferroelectrics 27, 191–194 (1980).
    [CrossRef]
  2. S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).
  3. W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, London, 1958).
  4. H. A. Wheeler, “Transmission Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. Microwave Theory Tech. MTT-13, 172–185 (1965).
    [CrossRef]
  5. K. J. Binns, P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems (Pergamon, New York, 1973).
  6. D. T. Paris, F. K. Hurd, Basic Electromagnetic Theory (McGraw-Hill, New York, 1969).
  7. G. E. Forsythe, W. R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960).
  8. R. V. Southwell, Relaxation Methods in Theoretical Physics, Vol. 1 (Oxford U. P., New York, 1946).
  9. W. T. Weeks, “Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,” IEEE Trans. Microwave Theory Tech. MTT-18, 35–43 (1970).
    [CrossRef]
  10. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1950).
  11. D. W. Kammler, “Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 925–937 (1968).
    [CrossRef]
  12. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 2.733.1 and 2.733.2 (defining u ≡ t − a and α ≡ (b2 − a2)1/2, thus t2 − 2at + b2 = u2 + a2).
  13. C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).
  14. H. Engen, “Excitation of Elastic Surface Waves by Spatial Harmonics of Interdigital Transducers,” IEEE Trans. Electron Devices ED-16, 1014–1017 (1969).
    [CrossRef]
  15. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), p. 257.
  16. S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
    [CrossRef]
  17. M. A. Title, L. M. Walpita, W. X. Chen, S. H. Lee, W. S. C. Chang, “Reactive Ion Beam Etching of PLZT Electrooptic Substrates with Repeated Self-Aligned Masking,” Appl. Opt. 25, 1508–1510 (1986).
    [CrossRef] [PubMed]
  18. D. R. Gabe, Principles of Metal Surface Treatment and Protection (Pergamon, Oxford, 1972).
  19. J. D. Greenwood, Heavy Deposition (Robert Draper, Teddington, U.K., 1970).
  20. F. A. Lowenheim, “Deposition of Inorganic Films from Solution,” in Thin Film Processes, J. L. Vossen, W. Kern, Eds. (Academic, New York, 1978).
  21. A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
    [CrossRef]
  22. K. Carl, K. Geisen, “Dielectric and Optical Properties of a Quasi-Ferroelectric PLZT Ceramic,” Proc. IEEE 61, 967–974 (1973).
    [CrossRef]

1987 (1)

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

1986 (2)

M. A. Title, L. M. Walpita, W. X. Chen, S. H. Lee, W. S. C. Chang, “Reactive Ion Beam Etching of PLZT Electrooptic Substrates with Repeated Self-Aligned Masking,” Appl. Opt. 25, 1508–1510 (1986).
[CrossRef] [PubMed]

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

1983 (1)

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

1980 (1)

J. D. Harris, V. Luft, “Electrode Slotting Process for Thin PLZT Wafers,” Ferroelectrics 27, 191–194 (1980).
[CrossRef]

1974 (1)

C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).

1973 (1)

K. Carl, K. Geisen, “Dielectric and Optical Properties of a Quasi-Ferroelectric PLZT Ceramic,” Proc. IEEE 61, 967–974 (1973).
[CrossRef]

1970 (1)

W. T. Weeks, “Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,” IEEE Trans. Microwave Theory Tech. MTT-18, 35–43 (1970).
[CrossRef]

1969 (1)

H. Engen, “Excitation of Elastic Surface Waves by Spatial Harmonics of Interdigital Transducers,” IEEE Trans. Electron Devices ED-16, 1014–1017 (1969).
[CrossRef]

1968 (1)

D. W. Kammler, “Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 925–937 (1968).
[CrossRef]

1965 (1)

H. A. Wheeler, “Transmission Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. Microwave Theory Tech. MTT-13, 172–185 (1965).
[CrossRef]

Binns, K. J.

K. J. Binns, P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems (Pergamon, New York, 1973).

Carl, K.

K. Carl, K. Geisen, “Dielectric and Optical Properties of a Quasi-Ferroelectric PLZT Ceramic,” Proc. IEEE 61, 967–974 (1973).
[CrossRef]

Chang, W. S. C.

Chen, W. X.

Drabik, T. J.

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

Engen, H.

H. Engen, “Excitation of Elastic Surface Waves by Spatial Harmonics of Interdigital Transducers,” IEEE Trans. Electron Devices ED-16, 1014–1017 (1969).
[CrossRef]

Esener, S.

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

Esener, S. C.

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

Forsythe, G. E.

G. E. Forsythe, W. R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

Gabe, D. R.

D. R. Gabe, Principles of Metal Surface Treatment and Protection (Pergamon, Oxford, 1972).

Geisen, K.

K. Carl, K. Geisen, “Dielectric and Optical Properties of a Quasi-Ferroelectric PLZT Ceramic,” Proc. IEEE 61, 967–974 (1973).
[CrossRef]

Gibbs, W. J.

W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, London, 1958).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 2.733.1 and 2.733.2 (defining u ≡ t − a and α ≡ (b2 − a2)1/2, thus t2 − 2at + b2 = u2 + a2).

Greenwood, J. D.

J. D. Greenwood, Heavy Deposition (Robert Draper, Teddington, U.K., 1970).

Haertling, G. H.

C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).

Harris, J. D.

J. D. Harris, V. Luft, “Electrode Slotting Process for Thin PLZT Wafers,” Ferroelectrics 27, 191–194 (1980).
[CrossRef]

Hegedds, Z.

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

Hurd, F. K.

D. T. Paris, F. K. Hurd, Basic Electromagnetic Theory (McGraw-Hill, New York, 1969).

Kammler, D. W.

D. W. Kammler, “Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 925–937 (1968).
[CrossRef]

Kertesz, L.

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

Kojnok, J.

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

Land, C. E.

C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).

Lawrenson, P. J.

K. J. Binns, P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems (Pergamon, New York, 1973).

Lee, S. H.

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

M. A. Title, L. M. Walpita, W. X. Chen, S. H. Lee, W. S. C. Chang, “Reactive Ion Beam Etching of PLZT Electrooptic Substrates with Repeated Self-Aligned Masking,” Appl. Opt. 25, 1508–1510 (1986).
[CrossRef] [PubMed]

Lowenheim, F. A.

F. A. Lowenheim, “Deposition of Inorganic Films from Solution,” in Thin Film Processes, J. L. Vossen, W. Kern, Eds. (Academic, New York, 1978).

Luft, V.

J. D. Harris, V. Luft, “Electrode Slotting Process for Thin PLZT Wafers,” Ferroelectrics 27, 191–194 (1980).
[CrossRef]

Paris, D. T.

D. T. Paris, F. K. Hurd, Basic Electromagnetic Theory (McGraw-Hill, New York, 1969).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 2.733.1 and 2.733.2 (defining u ≡ t − a and α ≡ (b2 − a2)1/2, thus t2 − 2at + b2 = u2 + a2).

Smythe, W. R.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1950).

Southwell, R. V.

R. V. Southwell, Relaxation Methods in Theoretical Physics, Vol. 1 (Oxford U. P., New York, 1946).

Szasz, A.

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

Thatcher, P. D.

C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).

Title, M.

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

Title, M. A.

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

M. A. Title, L. M. Walpita, W. X. Chen, S. H. Lee, W. S. C. Chang, “Reactive Ion Beam Etching of PLZT Electrooptic Substrates with Repeated Self-Aligned Masking,” Appl. Opt. 25, 1508–1510 (1986).
[CrossRef] [PubMed]

Walpita, L. M.

Wang, J. H.

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

Wasow, W. R.

G. E. Forsythe, W. R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

Weeks, W. T.

W. T. Weeks, “Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,” IEEE Trans. Microwave Theory Tech. MTT-18, 35–43 (1970).
[CrossRef]

Wheeler, H. A.

H. A. Wheeler, “Transmission Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. Microwave Theory Tech. MTT-13, 172–185 (1965).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), p. 257.

Appl. Opt. (1)

Appl. Solid State Sci. (1)

C. E. Land, P. D. Thatcher, G. H. Haertling, “Electrooptic Ceramics,” Appl. Solid State Sci. 4, 137–233 (1974).

Ferroelectrics (1)

J. D. Harris, V. Luft, “Electrode Slotting Process for Thin PLZT Wafers,” Ferroelectrics 27, 191–194 (1980).
[CrossRef]

IEEE Trans. Electron Devices (1)

H. Engen, “Excitation of Elastic Surface Waves by Spatial Harmonics of Interdigital Transducers,” IEEE Trans. Electron Devices ED-16, 1014–1017 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

D. W. Kammler, “Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 925–937 (1968).
[CrossRef]

W. T. Weeks, “Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,” IEEE Trans. Microwave Theory Tech. MTT-18, 35–43 (1970).
[CrossRef]

H. A. Wheeler, “Transmission Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. Microwave Theory Tech. MTT-13, 172–185 (1965).
[CrossRef]

J. Non-Cryst. Solids (1)

A. Szasz, J. Kojnok, L. Kertesz, Z. Hegedds, “On the Formation of Electroless Amorphous Layers,” J. Non-Cryst. Solids 57, 213–224 (1983).
[CrossRef]

Opt. Eng. (2)

S. C. Esener, J. H. Wang, T. J. Drabik, M. A. Title, S. H. Lee, “One-Dimensional Silicon/PLZT Spatial Light Modulators,” Opt. Eng. 26, 406–413 (1987).
[CrossRef]

S. H. Lee, S. Esener, M. Title, T. J. Drabik, “2-D Silicon/PLZT Spatial Light Modulators: Design Considerations and Technology,” Opt. Eng. 25, 250–260 (1986).

Proc. IEEE (1)

K. Carl, K. Geisen, “Dielectric and Optical Properties of a Quasi-Ferroelectric PLZT Ceramic,” Proc. IEEE 61, 967–974 (1973).
[CrossRef]

Other (11)

W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, London, 1958).

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1950).

K. J. Binns, P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems (Pergamon, New York, 1973).

D. T. Paris, F. K. Hurd, Basic Electromagnetic Theory (McGraw-Hill, New York, 1969).

G. E. Forsythe, W. R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

R. V. Southwell, Relaxation Methods in Theoretical Physics, Vol. 1 (Oxford U. P., New York, 1946).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 2.733.1 and 2.733.2 (defining u ≡ t − a and α ≡ (b2 − a2)1/2, thus t2 − 2at + b2 = u2 + a2).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), p. 257.

D. R. Gabe, Principles of Metal Surface Treatment and Protection (Pergamon, Oxford, 1972).

J. D. Greenwood, Heavy Deposition (Robert Draper, Teddington, U.K., 1970).

F. A. Lowenheim, “Deposition of Inorganic Films from Solution,” in Thin Film Processes, J. L. Vossen, W. Kern, Eds. (Academic, New York, 1978).

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Figures (18)

Fig. 1
Fig. 1

Comparison between geometries of transverse electrooptic modulation in (a) bulk electrooptic materials with those of planar structures with (b) surface or (c) embedded electrodes.

Fig. 2
Fig. 2

General definition of the problem to be studied: six electrodes of selected dimensions and separation are (a) set on or (b) embedded in an electrooptic material and switched with the voltages shown.

Fig. 3
Fig. 3

Discretization of the surface of surface electrode m.

Fig. 4
Fig. 4

Calculated electrode surface charge distributions for (a) surface electrodes 25 μm wide (on 50-μm centers) × 2.5 μm high and (b) embedded electrodes 10 μm wide (again 50-μm centers) × 15 μm deep. In both (a) and (b) we show the distribution of charge along the perimeter of electrode 1 (see Fig. 2) when the array modulating voltage V0 is 1 V. Charge density is given per unit length (μm) in the y-direction.

Fig. 5
Fig. 5

Calculated pseudocapacitance of electrode 1 in surface and embedded electrode arrays as a function of electrode geometry.

Fig. 6
Fig. 6

Electric field lines between electrodes 1 and 6 for the same geometries as in Fig. 4: (a) surface electrodes 25 μm wide (50-μm centers) × 2.5 μm high; (b) embedded electrodes 10 μm wide (50-μm centers) × 15 μm deep. Field line spacing along the vertical centerline is inversely proportional to the field magnitude, with the same proportionality in both (a) and (b).

Fig. 7
Fig. 7

Electric field magnitude and penetration (calculated down the array centerline) as a function of electrode geometry. Solid lines are for electrodes 10 μm wide (on 50-μm centers) and 0, 5, and 20 μm deep. Dashed lines are for electrodes 25 μm wide (on 50-μm centers) and 0, 5, and 20 μm deep.

Fig. 8
Fig. 8

PLZT optical index ellipsoid at a generalized location in the electrooptic material.

Fig. 9
Fig. 9

Calculated halfwave voltage at the center of the modulator array as a function of electrode geometry, for both reflective and transmissive EO modulators.

Fig. 10
Fig. 10

Calculated switching energy of alternative electrode geometries for reflective and transmissive EO modulators.

Fig. 11
Fig. 11

Modulated optical intensity profile between electrodes 1 and 6 as a function of electrode depth. Each curve is calculated with the appropriate halfwave voltage applied to the array; electrode width (10-μm) and separation (50-μm centers) are fixed.

Fig. 12
Fig. 12

Intensity modulation as a function of applied voltage. Solid line, integrated transmittance across the entire region between electrodes 1 and 6. Dotted line, transmittance at the center of the electrode array. Both profiles are calculated for surface electrodes 2.5 μm high, 25 μm wide on 50-μm centers.

Fig. 13
Fig. 13

Layout for the 16-bit bipolar 1-D Si/PLZT SLM.

Fig. 14
Fig. 14

Fabrication steps for embedded electrodes in polycrystalline PLZT.

Fig. 15
Fig. 15

Measured vs theoretical (calculated) capacitance of fabricated 1-D SLM electrode triplets.

Fig. 16
Fig. 16

(a) Polarization P and (b) birefringence Δn of 9/65/35 PLZT vs applied electric field E. While Δn ∝ P2 always, PE only for field magnitudes E < 10 kV/cm (1 V/μm). The slim-loop ferroelectric 9/65/35 PLZT also exhibits hysteresis.

Fig. 17
Fig. 17

Optical intensity modulation vs applied voltage for embedded electrodes 5 μm deep, 30 μm wide on 50-μm centers: curve (a) calculated for ideal 9/65/35 PLZT with quadratic Δn vs E; curve (b) calculated for 9/65/35 PLZT with saturated Δn vs E.

Fig. 18
Fig. 18

Measured intensity modulation vs applied voltage for (a) an unpoled device, (b) a device poled for 22 h at 400 V, and (c) a poled device after 12-h relaxation. The 9/65/35 PLZT modulator is fabricated using the same embedded electrode geometry as in Fig. 17.

Tables (1)

Tables Icon

Table I Green’s Function Coefficients A(z,z) and B(z,z) for Relative Positions of the Source Point (x,z) and Observation Point (x,z)

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

G ( x , z x , z ) = A ( z , z ) ln [ ( x - x ) 2 + ( z - z ) 2 ] + B ( z , z ) ln [ ( x - x ) 2 + ( z + z ) 2 ] ,
ϕ ( x , z ) = ϕ 0 + m = 1 M G [ x , z x m ( t ) , z m ( t ) ] w [ x m ( t ) , z m ( t ) ] d t ,
d m n 2 = ( x m , n + 1 - x m n ) 2 + ( z m , n + 1 - z m n ) 2 ,
tan θ m n = ( z m , n + 1 - z m n ) / ( x m , n + 1 - x m n ) .
ϕ ( x , z ) ϕ 0 + m = 1 M n = 1 N 0 d m n G ( x , z x m n + t cos θ m n , z m n + t sin θ m n ) × [ w m n + ( w m , n + 1 - w m n ) t / d m n ] d t ,
I ( 1 ) 0 d ( 1 - t / d ) ln ( t 2 - 2 a t + b 2 ) d t ,
I ( 2 ) 0 d ( t / d ) ln ( t 2 - 2 a t + b 2 ) d t .
ϕ ( x , z ) ϕ 0 + m = 1 M n = 1 N P m n ( x , z ) w m n ,
| a = a A ( x - x m n ) cos θ m n + ( z - z m n ) sin θ m n b 2 = b A 2 ( x - x m n ) 2 + ( z - z m n ) 2 d = d m n P m n ( x , z ) = A I ( 1 ) | a = a B ( x - x m n ) cos θ m n - ( z + z m n ) sin θ m n b 2 = b B 2 ( x - x m n ) 2 + ( z + z m n ) 2 d = d m n + B I ( 1 ) | a = a A ( x - x m , n - 1 ) cos θ m , n - 1 + ( z - z m , n - 1 ) sin θ m , n - 1 b 2 = b A 2 ( x - x m , n - 1 ) 2 + ( z - z m , n - 1 ) 2 d = d m , n - 1 + A I ( 2 ) | a = a B ( x - x m , n - 1 ) cos θ m , n - 1 + ( z + z m , n - 1 ) sin θ m , n - 1 b 2 = b B 2 ( x - x m , n - 1 ) 2 + ( z + z m , n - 1 ) 2 d = d m , n - 1 . + B I ( 2 )
ψ i = 1 M j = 1 N k = 1 K { ϕ ( x i j k , z i j k ) - [ ϕ 0 + m = 1 M n = 1 N P m n ( x i j k , z i j k ) w m n ] } 2 .
w ( M + 1 - m ) , n = - w m , n
P m n ( x , z ) P m n ( x , z ) - P ( M + 1 - m ) , n ( x , z )
Z 1 = i = 1 M j = 1 N k = 1 K P m n ( x i j k , z i j k ) P r s ( x i j k , z i j k ) ,
Z 2 = i = 1 M j = 1 N k = 1 K P r s ( x i j k , z i j k )
Z 4 = i = 1 M j = 1 N k = 1 K ϕ ( x i j k , z i j k ) P r s ( x i j k , z i j k ) for each r , s , = V 0 × i = 1 M / 2 j = 1 N k = 1 K P r s ( x i j k , z i j k ) ,
Z 5 = i = 1 M j = 1 N k = 1 K P m n ( x i j k , z i j k ) for each m , n ;
Z 6 = M × N × K ;
Z 8 = i = 1 M j = 1 N k = 1 K ϕ ( x i j k , z i j k ) , = V 0 × M / 2 × N × K ,
τ = C V π / I 0
E = ½ C V π 2 .
[ Q 1 Q 2 Q M ] = [ C 11 C 12 C 1 M C 21 C M 1 C M M ] × [ V 1 V 2 V M ] .
C m = Q m V 0 = 1 V 0 n = 1 N ( w m , n + w m , n + 1 ) 2 d m n ,
E x ( x , z ) = - ϕ ( x , z ) x = - m = 1 M n = 1 N [ x P m n ( x , z ) ] w m n ,
| a = a A b 2 = b A 2 d = d m n x P m n ( x , z ) = A I x ( 1 ) | a = a B b 2 = b B 2 d = d m n + B I x ( 1 ) | a = a A b 2 = b A 2 d = d m , n - 1 + A I x ( 2 ) | a = a B b 2 = b B 2 d = d m , n - 1 , + B I x ( 2 )
I x ( 1 ) I ( 1 ) x = I ( l ) a a x + I ( 1 ) ( b 2 ) ( b 2 ) x
E ( x , z ) = [ E x 2 ( x , z ) + E z 2 ( x , z ) ] 1 / 2 ,
Θ E ( x , z ) = tan - 1 [ E z ( x , z ) E x ( x , z ) ] .
n x ( x , z ) = [ cos 2 Θ E ( x , z ) ( n + n 3 2 R E 2 ( x , z ) ) 2 + sin 2 Θ E ( x , z ) n 2 ] - 1 / 2 ,
n y = n .
Γ ( x ) = 2 π m λ z = 0 l Δ n ( x , z ) d z
= 2 π m λ { z = 0 l [ cos 2 Θ E ( x , z ) ( n + n 3 2 R E 2 ( x , z ) ) 2 + sin 2 Θ E ( x , z ) n 2 ] - 1 / 2 d z - n l } ,
Γ ( x = 0 ) = π mn 3 R λ 0 l E 2 ( 0 , z ) d z .
Γ = π ( V V π ) 2 ,
V π = V 0 × [ mn 3 R λ z = 0 l E 0 2 ( 0 , z ) d z ] - 1 / 2 ,
T ( x ) = sin 2 [ Γ ( x ) 2 ] .
T ( x ) = sin 2 [ π m λ { z = 0 l [ cos 2 Θ E ( x , z ) ( n + n 3 2 R V 2 V 0 2 E 0 2 ( x , z ) ) 2 + sin 2 Θ E ( x , z ) n 2 ] - 1 / 2 d z - n l } ] .
sin 2 [ π 2 ( V V π ) 2 ]
T cell ( V ) = T ( x ) d x d x

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