Abstract

The problem of the grating action of a periodically distorted nematic liquid crystal layer, in the geometrical optics ray approximation is considered, and a theory for the calculation of the fringe powers is proposed. A nonabsorbing nematic phase is assumed, and the direction of incidence is taken to be normal to the layer. The powers of the resulting diffraction fringes are related to the spatial and angular deviation of the rays propagating across the layer and to the perturbation of the phase of the wave associated with the ray. The theory is applied to the simple case of a harmonically distorted nematic layer. In the case of a weakly distorted nematic layer the results agree with the predictions of Carroll’s model, where only even-order fringes are important. As the distortion becomes larger, odd-order fringes (with the exception of the first order) become equally important, and particularly those at relatively large orders (e.g., seven and nine) exhibit maxima greater than that of the even-order neighbors. Finally, the dependence of the powers of odd-order fringes on the distortion angle is quite different from that of the even-order fringes.

© 1987 Optical Society of America

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References

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  1. R. Williams, J. Chem. Phys. 39, 384 (1963).
    [Crossref]
  2. E. F. Carr, Adv. Chem. Ser. 63, 76 (1967).
    [Crossref]
  3. W. Helfrich, “Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations,” J. Chem. Phys. 51, 4092 (1969).
    [Crossref]
  4. E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
    [Crossref]
  5. P. A. Penz, G. W. Ford, “Electromagnetic Hydrodynamics of Liquid Crystals,” Phys. Rev. A 6, 414 (1972).
    [Crossref]
  6. R. Rigopoulos, H. M. Zenginoglou, “Electrohydrodynamic Instability Limits of Nematics,” Mol. Cryst. Liq. Cryst. 35, 307 (1976).
    [Crossref]
  7. H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
    [Crossref]
  8. H. M. Zenginoglou, I. Kosmopoulos, “On the Ability of Homogeneously Aligned Nematic Mesophases with Positive Dielectric Anisotropy to Exhibit Williams Domains as a Threshold Effect,” Mol. Cryst. Liq. Cryst. 43, 265 (1977).
    [Crossref]
  9. P. G. De Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).
  10. W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
    [Crossref]
  11. S. Lu, D. Jones, “Light Diffraction Phenomena in an a,c,-Excited Nematic Liquid Crystal Sample,” J. Appl. Phys. 42, 2138 (1971).
    [Crossref]
  12. T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
    [Crossref]
  13. R. A. Kashnow, J. E. Bigelow, “Diffraction from a Liquid Crystal Phase Grating,” Appl. Opt. 12, 2302 (1973).
    [Crossref] [PubMed]
  14. P. A. Penz, “Voltage-Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
    [Crossref]
  15. K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
    [Crossref]
  16. In “Optical Properties of Williams Domain,” S. Hirata, T. Tako, Jpn. J. of Appl. Phys. 21, 675 (1982), the ray path is calculated using a formula for the slope of the ray formula (3) in the article] which cannot be used unless the director field is nearly independent of x (Λ ≫ D). The results there were such that no focal lines are predicted.
    [Crossref]
  17. L. Landau, E. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1971), pp. 133 and 147.
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 130, 732, and 734.
  19. M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257 (1980).
    [Crossref]

1983 (1)

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

1982 (1)

In “Optical Properties of Williams Domain,” S. Hirata, T. Tako, Jpn. J. of Appl. Phys. 21, 675 (1982), the ray path is calculated using a formula for the slope of the ray formula (3) in the article] which cannot be used unless the director field is nearly independent of x (Λ ≫ D). The results there were such that no focal lines are predicted.
[Crossref]

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257 (1980).
[Crossref]

1977 (2)

H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
[Crossref]

H. M. Zenginoglou, I. Kosmopoulos, “On the Ability of Homogeneously Aligned Nematic Mesophases with Positive Dielectric Anisotropy to Exhibit Williams Domains as a Threshold Effect,” Mol. Cryst. Liq. Cryst. 43, 265 (1977).
[Crossref]

1976 (1)

R. Rigopoulos, H. M. Zenginoglou, “Electrohydrodynamic Instability Limits of Nematics,” Mol. Cryst. Liq. Cryst. 35, 307 (1976).
[Crossref]

1973 (1)

1972 (2)

P. A. Penz, G. W. Ford, “Electromagnetic Hydrodynamics of Liquid Crystals,” Phys. Rev. A 6, 414 (1972).
[Crossref]

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

1971 (3)

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

S. Lu, D. Jones, “Light Diffraction Phenomena in an a,c,-Excited Nematic Liquid Crystal Sample,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
[Crossref]

1970 (1)

P. A. Penz, “Voltage-Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

1969 (1)

W. Helfrich, “Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations,” J. Chem. Phys. 51, 4092 (1969).
[Crossref]

1967 (1)

E. F. Carr, Adv. Chem. Ser. 63, 76 (1967).
[Crossref]

1963 (1)

R. Williams, J. Chem. Phys. 39, 384 (1963).
[Crossref]

Arakawa, M.

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257 (1980).
[Crossref]

Bigelow, J. E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 130, 732, and 734.

Carr, E. F.

E. F. Carr, Adv. Chem. Ser. 63, 76 (1967).
[Crossref]

Carroll, T. O.

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

De Gennes, P. G.

E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
[Crossref]

P. G. De Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

Dubois-Violette, E.

E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
[Crossref]

Ford, G. W.

P. A. Penz, G. W. Ford, “Electromagnetic Hydrodynamics of Liquid Crystals,” Phys. Rev. A 6, 414 (1972).
[Crossref]

Fukuda, A.

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

Greubel, W.

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

Helfrich, W.

W. Helfrich, “Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations,” J. Chem. Phys. 51, 4092 (1969).
[Crossref]

Hirata, S.

In “Optical Properties of Williams Domain,” S. Hirata, T. Tako, Jpn. J. of Appl. Phys. 21, 675 (1982), the ray path is calculated using a formula for the slope of the ray formula (3) in the article] which cannot be used unless the director field is nearly independent of x (Λ ≫ D). The results there were such that no focal lines are predicted.
[Crossref]

Jones, D.

S. Lu, D. Jones, “Light Diffraction Phenomena in an a,c,-Excited Nematic Liquid Crystal Sample,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

Kashnow, R. A.

Kondo, K.

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

Kosmopoulos, I.

H. M. Zenginoglou, I. Kosmopoulos, “On the Ability of Homogeneously Aligned Nematic Mesophases with Positive Dielectric Anisotropy to Exhibit Williams Domains as a Threshold Effect,” Mol. Cryst. Liq. Cryst. 43, 265 (1977).
[Crossref]

H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
[Crossref]

Kuze, E.

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

Landau, L.

L. Landau, E. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1971), pp. 133 and 147.

Lifshitz, E.

L. Landau, E. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1971), pp. 133 and 147.

Lu, S.

S. Lu, D. Jones, “Light Diffraction Phenomena in an a,c,-Excited Nematic Liquid Crystal Sample,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

Parodi, O.

E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
[Crossref]

Penz, P. A.

P. A. Penz, G. W. Ford, “Electromagnetic Hydrodynamics of Liquid Crystals,” Phys. Rev. A 6, 414 (1972).
[Crossref]

P. A. Penz, “Voltage-Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

Rigopoulos, R.

H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
[Crossref]

R. Rigopoulos, H. M. Zenginoglou, “Electrohydrodynamic Instability Limits of Nematics,” Mol. Cryst. Liq. Cryst. 35, 307 (1976).
[Crossref]

Tako, T.

In “Optical Properties of Williams Domain,” S. Hirata, T. Tako, Jpn. J. of Appl. Phys. 21, 675 (1982), the ray path is calculated using a formula for the slope of the ray formula (3) in the article] which cannot be used unless the director field is nearly independent of x (Λ ≫ D). The results there were such that no focal lines are predicted.
[Crossref]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257 (1980).
[Crossref]

Williams, R.

R. Williams, J. Chem. Phys. 39, 384 (1963).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 130, 732, and 734.

Wolf, H.

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

Zenginoglou, H. M.

H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
[Crossref]

H. M. Zenginoglou, I. Kosmopoulos, “On the Ability of Homogeneously Aligned Nematic Mesophases with Positive Dielectric Anisotropy to Exhibit Williams Domains as a Threshold Effect,” Mol. Cryst. Liq. Cryst. 43, 265 (1977).
[Crossref]

R. Rigopoulos, H. M. Zenginoglou, “Electrohydrodynamic Instability Limits of Nematics,” Mol. Cryst. Liq. Cryst. 35, 307 (1976).
[Crossref]

Adv. Chem. Ser. (1)

E. F. Carr, Adv. Chem. Ser. 63, 76 (1967).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

J. Appl. Phys. (2)

S. Lu, D. Jones, “Light Diffraction Phenomena in an a,c,-Excited Nematic Liquid Crystal Sample,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

J. Chem. Phys. (2)

W. Helfrich, “Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations,” J. Chem. Phys. 51, 4092 (1969).
[Crossref]

R. Williams, J. Chem. Phys. 39, 384 (1963).
[Crossref]

J. Phys. (1)

E. Dubois-Violette, P. G. De Gennes, O. Parodi, “Hydrodynamic Instabilities of Nematic Liquid Crystals under a,c, Electric Fields,” J. Phys. 32, 305 (1971).
[Crossref]

Jpn. J. Appl. Phys. (1)

K. Kondo, M. Arakawa, A. Fukuda, E. Kuze, “Light Propagation in Williams Domains as Analysed Numerically by Geometrical Optics.” Jpn. J. Appl. Phys. 22, 394 (1983).
[Crossref]

Jpn. J. of Appl. Phys. (1)

In “Optical Properties of Williams Domain,” S. Hirata, T. Tako, Jpn. J. of Appl. Phys. 21, 675 (1982), the ray path is calculated using a formula for the slope of the ray formula (3) in the article] which cannot be used unless the director field is nearly independent of x (Λ ≫ D). The results there were such that no focal lines are predicted.
[Crossref]

Mol. Cryst. Liq. Cryst. (3)

R. Rigopoulos, H. M. Zenginoglou, “Electrohydrodynamic Instability Limits of Nematics,” Mol. Cryst. Liq. Cryst. 35, 307 (1976).
[Crossref]

H. M. Zenginoglou, R. Rigopoulos, I. Kosmopoulos, “On the Electrohydrodynamic Instability Limits of Nematics under the Action of Sinusoidal Electric Fields,” Mol. Cryst. Liq. Cryst. 39, 27 (1977).
[Crossref]

H. M. Zenginoglou, I. Kosmopoulos, “On the Ability of Homogeneously Aligned Nematic Mesophases with Positive Dielectric Anisotropy to Exhibit Williams Domains as a Threshold Effect,” Mol. Cryst. Liq. Cryst. 43, 265 (1977).
[Crossref]

Phys. Rev. A (1)

P. A. Penz, G. W. Ford, “Electromagnetic Hydrodynamics of Liquid Crystals,” Phys. Rev. A 6, 414 (1972).
[Crossref]

Phys. Rev. Lett. (1)

P. A. Penz, “Voltage-Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257 (1980).
[Crossref]

Other (3)

L. Landau, E. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1971), pp. 133 and 147.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 130, 732, and 734.

P. G. De Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Defining directions and angles entering the theory.

Fig. 3
Fig. 3

Paths of two neighboring rays and the angle formed by them with they axis at the various stages of their propagation. For details see text.

Fig. 4
Fig. 4

Illustration of the quantities entering the diffraction integral.

Fig. 5
Fig. 5

Ray paths inside the NLC layer when the deformation angle is (a) ϑm = 15°, (b) ϑm = 30°, and (c) ϑm = 45°. As it is seen at ϑm = 30°, intersections of rays occur and consequently caustic surfaces begin to form. Nevertheless the ray path calculations at deformation angles greater than this value are useful in that they help greatly in explaining qualitatively what one sees when observing the distorted layer in a microscope.

Fig. 6
Fig. 6

Angular deviation of the transmitted rays vs the x coordinate of the point of entrance at (a) ϑm = 10°, (b) ϑm = 20°, and (c) ϑm = 30°. As explained in the text, the effect of Ot is corrective only.

Fig. 7
Fig. 7

Spatial deviation ΔxD of the transmitted rays vs the x coordinate of the point of entrance at (a) ϑm = 10°, (b) ϑm = 20°, and (c) ϑm = 30°. To help comparison with the phase deviation ΔΦt, ΔxD is multiplied with the spatial distortion wavenumber q.

Fig. 8
Fig. 8

Perturbation of the phase of the wave associated with the rays as a function of the x coordinate of the entrance point at (a) ϑm = 10°, (b) ϑm = 20°, and (c) ϑm = 30°. On the average this perturbation is an order of magnitude greater than qΔxD and, thus, at small ϑm values is the sole mechanism for the fringe formation leading to Carroll’s approximation.

Fig. 9
Fig. 9

Fringe powers of the central and the first three even-order fringes as functions of the deformation angle ϑm.

Fig. 10
Fig. 10

Fringe powers of the first five odd-order fringes as functions of the deformation angle θm. The power scale used here is amplified by a factor equal to 10 compared with that of Fig. 9. Nevertheless, with the exception of the first order, the powers of odd-order fringes exhibit maxima comparable with and, in some cases, greater than the maxima of even-order fringes.

Equations (34)

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n x = cos ϑ , n y = sin ϑ , n z = 0 , ϑ = ϑ ( x , y ) = θ ( x + m Λ , y ) ,
k = n e k 0 ( 1 + β sin 2 ψ ) - 1 / 2 ,
S = 1 8 π c n e E 2 ( 1 - α sin 2 ϕ ) 1 / 2 ,
tan ϕ = ( n e n o ) 2 tan ψ ,
β ( n e / n o ) 2 - 1 , α 1 - ( n o / n e ) 2 = β / ( 1 + β ) ,
L 1 k 0 ( x 0 , 0 , z 0 ) ( x D , D , z 0 ) k · d l ,
L n e 0 D d y [ ( 1 - α sin 2 ϕ ) 1 / 2 / cos θ ] ,
ϕ = θ + ϑ and tan θ = d x d y .
d 2 x d y 2 = 1 cos 2 θ f 3 1 - α { [ ( f - 1 - α f 3 ) tan θ - α sin 2 ϕ 2 f ] ϑ x + [ f - 1 - α f 3 + α sin 2 ϕ 2 f tan θ ] ϑ y } ,
tan θ 0 = β tan ϑ 0 1 + ( n e n o ) 2 tan 2 ϑ 0
S 0 cos θ 0 d x 0 = S D cos θ D d x D ,
E D = E 0 ( f f D cos θ 0 cos θ D ) 1 / 2 | d x D d x 0 | - 1 / 2 ,
f 0 ( 1 - α sin 2 ϕ 0 ) 1 / 2 , ϕ 0 θ 0 + ϑ 0 , f D ( 1 - α sin 2 ϕ D ) 1 / 2 , ϕ D θ D + ϑ D ,
sin θ t = n e ( n o / n e ) 2 tan ϕ D cos ϑ D - sin ϑ D [ 1 + ( n o n e tan ϕ D ) 2 ] 1 / 2 ,
tan ( θ D - ϑ D ) = n e n o - n o n e cos ϑ D sin ϑ D + sin θ t n o 2 cos 2 ϑ D + n e 2 sin 2 ϑ D - sin 2 θ t n o 2 cos 2 ϑ D - sin 2 θ t ,
E t = 2 n e ( n e + 1 ) 2 H | d x D d x 0 | - 1 / 2 E i ,
H ( cos θ 0 cos θ D f 0 f D ) 1 / 2 ( cos θ D cos θ D + f D f D ) ( cos θ 0 + n e f 0 ) ( cos θ t cos θ D + 1 n e f D ) · ( n e + 1 ) 2 2 n e ,
E f = A 0 L X 0 L z d x D d z D E t exp ( - j Φ t ) exp ( - j k 0 r ) r cos θ t + cos γ 2 ,
E f = A exp ( - j k 0 r ) r L z sin ( k 0 L z sin γ sin ɛ / 2 ) k 0 L z sin γ sin ɛ / 2 × 0 L x d x D E t exp [ j ( k 0 x D sin γ cos ɛ - Φ t ) ] cos θ t + cos γ 2 ,
E f = A exp ( - j k 0 r ) r L x L z G 1 · G 2 · F ,
G 1 sin ( k 0 L z sin γ sin ɛ / 2 ) k 0 L z sin γ sin ɛ / 2 ,
G 2 sin ( N k 0 Λ sin γ cos ɛ / 2 ) N sin ( k 0 Λ sin γ cos ɛ / 2 ,
F 1 Λ 0 Λ d x 0 | d x D d x 0 | 1 / 2 E t · exp [ j ( k 0 x D sin γ cos ɛ - Φ t ) ] cos θ t + cos γ 2 } ,
P n = 1 cos γ n | 1 Λ 0 Λ d x 0 | 1 + d Δ x D d x 0 | 1 / 2 · H 1 2 ( cos γ n + cos θ t ) exp [ j ( n q x 0 + n q Δ x D - Δ Φ t ) ] 2 ,
ϑ = ϑ m sin q x sin ( π y / D ) .
P 2 n J n ( A 2 ϑ m 2 ) 2 , P 2 n + 1 0 ,
Π Ω = c 8 π Ω d Ω r 2 E f 2 .
Π n = c A 2 L x 2 L 2 2 8 π d γ sin γ d ɛ G 1 2 ( γ , ɛ ) D 2 2 ( γ , ɛ ) F ( γ , ɛ ) 2 .
Π n = c A 2 L x 2 L z 2 8 π k 0 d k x d k z k 0 2 - k x 2 - k z 2 G 1 2 ( k z ) G 2 2 ( k x ) F ( k x ) 2 .
Π n = c A 2 L x 2 L z 2 8 π k 0 d k x G 2 2 ( k x ) F ( k x ) 2 k 0 2 - k x 2 d k z G 1 2 ( k z ) .
- 2 m π N Λ < Δ k x < 2 m π N Λ ,
Π n = π c A 2 L x L z 8 k 0 2 F ( n q ) 2 cos γ n .
Π 0 ( 0 ) = π c A 2 L x L z 8 k 0 F ( 0 ) ( 0 ) 2 ,
F ( 0 ) ( 0 ) = 2 n e ( n e + 1 ) 2 E i exp ( - j k 0 n e D ) .

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