Abstract

A general analytical formula describing the transfer function between the input and output of an arbitrary rectangular multimode interference (MMI) coupler has been derived using the elliptic theta function ϑ(x′, z′). This formula provides the positions, amplitudes and relative phases of all the self-images of a given source. It is shown how the transfer function can be used as a propagation matrix for any rectangular NxM MMI. Specific simplified solutions for NxN, symmetric and paired MMIs are also derived from the general formula.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  12. D.C. Chang and E.F. Kuester, “A Hybrid Method for Paraxial Beam Propagation in Multimode Optical Waveguides,” Microwave Theory and Techniques, IEEE Transactions on,  29(9), 923–933 (1981).
    [Crossref]
  13. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
    [Crossref]
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    [Crossref]
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    [Crossref]
  25. X. Ouyang, C. Antony, F. Gunning, H. Zhang, and Y. L. Guan, “Discrete Fresnel Transform and Its Circular Convolution,” http://arxiv.org/abs/1510.00574 .
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    [Crossref]

2013 (1)

2010 (1)

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

2009 (1)

1999 (1)

1997 (1)

D. L. Aronstein and C. R. Stroud, “Fractional wave-function revivals in the infinite square well,” Phys. Rev. A 55(6), 4526–4537 (1997).
[Crossref]

1996 (3)

V. Arrizón, J. G. Ibarra, and J. Ojeda-Castañeda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13(12), 2414–2422 (1996).
[Crossref]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

M. V. Berry, “Quantum fractals in boxes,” J. Phys. A: Math. Gen.,  29(20), 6617–6629 (1996).
[Crossref]

1995 (3)

1994 (1)

1993 (1)

1981 (1)

D.C. Chang and E.F. Kuester, “A Hybrid Method for Paraxial Beam Propagation in Multimode Optical Waveguides,” Microwave Theory and Techniques, IEEE Transactions on,  29(9), 923–933 (1981).
[Crossref]

1975 (1)

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27(6), 337–339 (1975).
[Crossref]

1973 (1)

1968 (1)

L. A. Rivlin and V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Zzv. VUZ Radiofizika 11(4), 572–578 (1968).

1965 (1)

1959 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, “Fourier Images,” Proc. Phys. Soc. B 70, 486 (1957).
[Crossref]

1881 (1)

L. Rayleigh, “On Copying Diffraction Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 403–405 (1836).

Ankele, G.

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27(6), 337–339 (1975).
[Crossref]

Arndt, M.

Aronstein, D. L.

D. L. Aronstein and C. R. Stroud, “Fractional wave-function revivals in the infinite square well,” Phys. Rev. A 55(6), 4526–4537 (1997).
[Crossref]

Arrizon, V.

Arrizón, V.

Bachmann, M.

Berry, M. V.

M. V. Berry, “Quantum fractals in boxes,” J. Phys. A: Math. Gen.,  29(20), 6617–6629 (1996).
[Crossref]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum Carpets, Carpets of Light,” Phys. World16 (6), (2001).

Besse, P. A.

Breazeale, M. A.

Bryngdahl, O.

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972)

Case, W.B.

Chang, D.C.

D.C. Chang and E.F. Kuester, “A Hybrid Method for Paraxial Beam Propagation in Multimode Optical Waveguides,” Microwave Theory and Techniques, IEEE Transactions on,  29(9), 923–933 (1981).
[Crossref]

Chen, R.T.

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier Images,” Proc. Phys. Soc. B 70, 486 (1957).
[Crossref]

Deachapunya, S.

Hiedemann, E. A.

Hosseini, A.

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Ibarra, J. G.

Ireland, K.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer-Verlag, 1990).
[Crossref]

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972)

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

Kuester, E.F.

D.C. Chang and E.F. Kuester, “A Hybrid Method for Paraxial Beam Propagation in Multimode Optical Waveguides,” Microwave Theory and Techniques, IEEE Transactions on,  29(9), 923–933 (1981).
[Crossref]

Kwong, D.

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Lee, B.S.

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Lin, C.

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Marzoli, I.

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum Carpets, Carpets of Light,” Phys. World16 (6), (2001).

Melchior, H.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier Images,” Proc. Phys. Soc. B 70, 486 (1957).
[Crossref]

Ojeda-Castañeda, J.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: Principles and applications,” IEEE J. Light. Tech. 13(4), 615–627 (1995).
[Crossref]

Rayleigh, L.

L. Rayleigh, “On Copying Diffraction Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Rivlin, L. A.

L. A. Rivlin and V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Zzv. VUZ Radiofizika 11(4), 572–578 (1968).

Rosen, M.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer-Verlag, 1990).
[Crossref]

Schleich, W. P.

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum Carpets, Carpets of Light,” Phys. World16 (6), (2001).

Shul’dyaev, V. S.

L. A. Rivlin and V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Zzv. VUZ Radiofizika 11(4), 572–578 (1968).

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: Principles and applications,” IEEE J. Light. Tech. 13(4), 615–627 (1995).
[Crossref]

Stankovic, S.

Stroud, C. R.

D. L. Aronstein and C. R. Stroud, “Fractional wave-function revivals in the infinite square well,” Phys. Rev. A 55(6), 4526–4537 (1997).
[Crossref]

Szwaykowski, P.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 403–405 (1836).

Tomandl, M.

Tschudi, T.

Ulrich, R.

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27(6), 337–339 (1975).
[Crossref]

Wen, J.

Winthrop, J. T.

Worthington, c. R.

Xiao, M.

Zhang, Y.

Zhou, C.

Adv. Opt. Photon. (1)

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27(6), 337–339 (1975).
[Crossref]

IEEE J. Light. Tech. (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: Principles and applications,” IEEE J. Light. Tech. 13(4), 615–627 (1995).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Phys. A: Math. Gen. (1)

M. V. Berry, “Quantum fractals in boxes,” J. Phys. A: Math. Gen.,  29(20), 6617–6629 (1996).
[Crossref]

Microwave Theory and Techniques, IEEE Transactions on (1)

D.C. Chang and E.F. Kuester, “A Hybrid Method for Paraxial Beam Propagation in Multimode Optical Waveguides,” Microwave Theory and Techniques, IEEE Transactions on,  29(9), 923–933 (1981).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 403–405 (1836).

L. Rayleigh, “On Copying Diffraction Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Phys. Rev. A (1)

D. L. Aronstein and C. R. Stroud, “Fractional wave-function revivals in the infinite square well,” Phys. Rev. A 55(6), 4526–4537 (1997).
[Crossref]

Proc. Phys. Soc. B (1)

J. M. Cowley and A. F. Moodie, “Fourier Images,” Proc. Phys. Soc. B 70, 486 (1957).
[Crossref]

Selected Topics in Quantum Electronics, IEEE Journal of (1)

A. Hosseini, D. Kwong, C. Lin, B.S. Lee, and R.T. Chen, “Output Formulation for Symmetrically Excited One-to-N Multimode Interference Coupler,” Selected Topics in Quantum Electronics, IEEE Journal of,  16(1), 61–69, (2010).
[Crossref]

Zzv. VUZ Radiofizika (1)

L. A. Rivlin and V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Zzv. VUZ Radiofizika 11(4), 572–578 (1968).

Other (4)

X. Ouyang, C. Antony, F. Gunning, H. Zhang, and Y. L. Guan, “Discrete Fresnel Transform and Its Circular Convolution,” http://arxiv.org/abs/1510.00574 .

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer-Verlag, 1990).
[Crossref]

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972)

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum Carpets, Carpets of Light,” Phys. World16 (6), (2001).

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Figures (22)

Fig. 1
Fig. 1 Construction of the extended field.
Fig. 2
Fig. 2 The images of a single source located at x0.
Fig. 3
Fig. 3 A single image.
Fig. 4
Fig. 4 The two interpretations of the single image at z = L.
Fig. 5
Fig. 5 Two images at z′ = 1/2.
Fig. 6
Fig. 6 Two of the four images destructively interfere.
Fig. 7
Fig. 7 Propagation in a 2D MMI with W x = 3 / 2 W y.
Fig. 8
Fig. 8 A single image at z = L/4 due to symmetric interference.
Fig. 9
Fig. 9 A schematic of symmetric interference.
Fig. 10
Fig. 10 Symmetric interference producing 3 images.
Fig. 11
Fig. 11 Paired interference producing a single image at z = L/3.
Fig. 12
Fig. 12 Graphical representation of equation (51).
Fig. 13
Fig. 13 The two outputs of one input for a paired 2 × 2 MMI.
Fig. 14
Fig. 14 The four outputs of one input for a paired 2 × 4 MMI.
Fig. 15
Fig. 15 Schematic of a general N × N MMI.
Fig. 16
Fig. 16 s = 1 output for a regular 4 × 4 MMI.
Fig. 17
Fig. 17 s = 2 output for a regular 4 × 4 MMI.
Fig. 18
Fig. 18 Diagram for a restricted MMI.
Fig. 19
Fig. 19 s = 1 output for a restricted 4 × 4 MMI.
Fig. 20
Fig. 20 s = 2 output for a restricted 4 × 4 MMI.
Fig. 21
Fig. 21 The rectangular approximation for q = 7.
Fig. 22
Fig. 22 A simulation of lightwave propagation in a waveguide using MMI matrices.

Tables (1)

Tables Icon

Table 1 Properties of the MMI matrices derived from ϑ(x′, z′)

Equations (127)

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d 2 E y ( x ) d x 2 + ( k 2 n ( x ) 2 β 2 ) E y ( x ) = 0
E y ( x ) = m = 0 a m ψ m ( x )
β m = k n r 1 ( m π W e k n r ) 2
β m k n r 1 2 k n r ( m π W e ) 2
E y ( x , z ) = m = 0 a m sin ( m π x W e ) e i β m z e i k n r z m = 0 m = a m sin ( m π x W e ) e i 2 k n r ( m π W e ) 2 z
F ( x , z ) = { E y ( x , z ) 0 x W e E y ( 2 W e x , z ) W e x 2 W e
F ( x , z ) = m = a m e i m π x W e e i β m z e i k n r z m = 0 a m e i m π x W e e i 2 k n r ( m π W e ) 2 z
F ( x , z ) = e i k n r z L m = a m e im 2 π x e im 2 π z
ϑ ( x , z ) = m = e im 2 π x e im 2 π z
F ( x , 0 ) * ϑ ( x , z ) = n = a n e i n 2 π x * m = e im 2 π x e im 2 π z = n , m a n e im 2 π z ( e i n 2 π x * e im 2 π x ) = m = a m e im 2 π x e im 2 π z
F ( x , z ) = e i k n r z L F ( x , 0 ) * ϑ ( x , z )
ϑ ( x , z + 2 ) = m = e im 2 π x e im 2 π ( z + 2 ) = m = e im 2 π x e im 2 π z = ϑ ( x , z )
ϑ ( x , 0 ) = m = e im 2 π x = m = δ ( x m ) = δ 1 ( x )
ϑ ( x , 1 q ) = 1 q t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) δ ( x t 2 q )
F ( x , 1 q ) = e i k n r L q q t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) F ( x t 2 q , 0 )
F ( x , 1 q ) = e i k n r L q q t 2 q t q ( mod 2 ) e i π 4 ( 1 t 2 q ) F ( x t 2 q , 0 )
E ( x , 1 q ) = e i k n r L q q t = 0 t q ( mod 2 ) 2 q 1 [ e i π 4 ( 1 t 2 q ) E y ( x t 2 q , 0 ) e i π 4 ( 1 t 2 q ) E y ( x t 2 q , 0 ) ]
E y ( x , 0 ) = j = 1 N A j E 0 ( x x j ) = j = 1 N A j E 0 ( x ) * δ ( x x j ) = E 0 ( x ) * ( j = 1 N A j δ ( x x j ) )
F ( x , 0 ) = j = 1 N A j ( E 0 ( x x j ) E 0 ( x + x j ) ) = j = 1 N A j ( E 0 ( x ) * δ ( x x j ) E 0 ( x ) * δ ( x + x j ) )
F ( x , 0 ) = j = 1 N A j ( E 0 ( x ) * δ ( x x j ) E 0 ( x ) * δ ( x + x j ) ) = E 0 ( x , 0 ) * ( j = 1 N A j ( δ ( x x j ) δ ( x + x j ) ) )
F ( x , 0 ) = j = 1 N A j ( E 0 ( x ) * δ ( x x j ) + E 0 ( x ) * δ ( x + x j ) ) = E 0 ( x , 0 ) * ( j = 1 N A j ( δ ( x x j ) + δ ( x + x j ) ) )
F ( x , 1 q ) = ( E 0 ( x , 0 ) * ( j = 1 N A j ( δ ( x x j ) δ ( x + x j ) ) ) ) * ( e i k n r L q q t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) δ ( x t 2 q ) )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( ( j = 1 N A j ( δ ( x x j ) δ ( x + x j ) ) ) * ( t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) δ ( x t 2 q ) ) ) = e i k n r L q q E 0 ( x ) * ( j = 1 N t = 0 t q ( mod 2 ) 2 q 1 A j e i π 4 ( 1 t 2 q ) ( δ ( x x j ) δ ( x + x j ) ) * δ ( x t 2 q ) ) = e i k n r L q q E 0 ( x ) * ( j = 1 N t = 0 t q ( mod 2 ) 2 q 1 A j e i π 4 ( 1 t 2 q ) ( δ ( x x j t 2 q ) δ ( x + x j = t 2 q ) ) )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) ( δ ( x x 1 t 2 q ) δ ( x + x 1 t 2 q ) ) )
F ( x , z = 1 1 ) = e i k n r L E 0 ( x ) * ( δ ( x x 1 1 2 ) δ ( x + x 1 1 2 ) )
E y ( x , z = 1 ) = e i k n r L E 0 ( x ) * δ ( x + x 1 1 2 )
F ( x , 1 2 ) = e i k n r L 2 2 E 0 ( x ) * ( t = 0 t 2 ( mod 2 ) 3 e i π 4 ( 1 t 2 2 ) ( δ ( x x 1 t 4 ) δ ( x + x 1 t 4 ) ) ) = e i k n r L 2 2 E 0 ( x ) * ( e i π 4 ( δ ( x x 1 ) δ ( x + x 1 ) ) + e i π 4 ( δ ( x x 1 1 2 ) δ ( x + x 1 1 2 ) ) )
E y ( x , 1 2 ) = e i k n r L 2 2 E 0 ( x ) * ( e i π 4 δ ( x x 1 ) e i π 4 δ ( x + x 1 1 2 ) )
F ( x , 1 4 ) = e i k n r L 4 2 E 0 ( x ) * ( t = 0 t 4 ( mod 2 ) 7 e i π 4 ( 1 t 2 4 ) ( δ ( x 1 8 t 8 ) δ ( x + 1 8 t 8 ) ) ) = e i k n r L 4 2 E 0 ( x ) * ( e i π 4 ( δ ( x 1 8 ) δ ( x + 1 8 ) ) + ( δ ( x 3 8 ) δ ( x 1 8 ) ) + e i 3 π 4 ( δ ( x 5 8 ) δ ( x 3 8 ) ) + ( δ ( x 7 8 ) δ ( x 5 8 ) ) )
E y ( x , 1 4 ) = e i k n r L 4 2 E 0 ( x ) * ( ( e i π 4 1 ) δ ( x 1 8 ) + ( 1 e i 3 π 4 ) δ ( x 3 8 ) ) = e i k n r L 4 E 0 ( x ) * ( 1 2 ( 1 1 2 ) e i 5 π 8 δ ( x 1 8 ) + 1 2 ( 1 + 1 2 ) e i π 8 δ ( x 3 8 ) )
ψ m , n ( x , y ) = sin ( m π x W x ) sin ( n π y W y )
β m , n = n r 2 k 2 ( m π W x ) 2 ( n π W y ) 2 = n r k 1 ( m π n r k W x ) 2 ( n π n r k W y ) 2 n r k ( 1 1 2 ( m π n r k W x ) 2 1 2 ( n π n r k W y ) 2 ) = n r k m 2 π L x n 2 π L y
E ( x , y , z ) = m , n a m , n ψ m , n ( x , y ) e i β m , n z e i k n r z m , n a m , n sin ( m π x W x ) sin ( n π y W y ) e im 2 π z L x + i n 2 π z L y
E ( x , y , z ) = e i k n r z ( m b m sin ( m π x W x ) e im 2 π z L x ) ( n c n sin ( n π y W y ) e im 2 π z L y )
F ( x , 0 ) = m = m = a m e im 2 π x
F ( x , 0 ) = m odd a m e im 2 π x
F ( x , z ) = e i k n r z L m odd a m e im 2 π x e im 2 π z = e i k n r z L F ( x , 0 ) * ϑ odd ( x , z )
ϑ odd ( x , z ) = m odd e im 2 π x e im 2 π z
F ( x , 1 q ) = F ( x , 1 4 q ) e i k n r L q 4 q E 0 ( x ) * ( t = 0 t 4 q ( mod 2 ) 8 q 1 e i π 4 ( 1 t 2 4 q ) ( δ ( x 1 4 t 8 q ) δ ( x + 1 4 t 8 q ) ) )
F ( x , 1 q ) = F ( x , 1 4 q ) = e i k n r L q 4 q E 0 ( x ) * ( t = 0 4 q 1 e i π 4 ( 1 t 2 q ) ( δ ( x 1 4 t 4 q ) δ ( x + 1 4 t 4 q ) ) )
F ( x , 1 4 q ) = e i k n r L q 4 q E 0 ( x ) * ( t = 0 4 q 1 e i π 4 ( 1 t 2 q ) δ ( x 1 4 t 4 q ) e i π 4 ( 1 ( t + 2 q ) 2 q ) δ ( x + 1 4 t + 2 q 4 q ) ) = e i k n r L q 4 q E 0 ( x ) * ( t = 0 4 q 1 ( e i π 4 ( 1 t 2 q ) e i π 4 ( 1 ( t + 2 q ) 2 q ) ) δ ( x 1 4 t 4 q ) )
e i π 4 ( 1 t 2 q ) e i π 4 ( 1 ( t + 2 q ) 2 q ) = e i π 4 ( 1 t 2 q ) e i π 4 ( 1 t 2 + 4 t q + 4 q 2 q ) = e i π 4 ( 1 t 2 q ) ( 1 e i π ( t + q ) )
F ( x , 1 4 q ) = e i k n r L q q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 4 q 1 e i π 4 ( 1 t 2 q ) δ ( x 1 4 t 4 q ) )
F ( x , 1 4 q ) = e i k n r L q q E 0 ( x ) * ( t = 2 q t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) δ ( x 1 4 t 4 q ) )
E y ( x , 1 4 q ) = e i k n r L q q E 0 ( x ) * ( t = q + 1 t q ( mod 2 ) q 1 e i π 4 ( 1 t 2 q ) δ ( x 1 4 t 4 q ) )
E y ( x , 1 12 ) = e i k n r L 12 3 E 0 ( x , 0 ) * ( t = 2 t even 2 e i π 4 ( 1 t 2 3 ) δ ( x 1 4 t 12 ) ) = e i k n r L 12 3 E 0 ( x , 0 ) * ( e i π 4 ( 1 4 3 ) δ ( x 1 4 + 1 6 ) + e i π 4 δ ( x 1 4 ) + e i π 4 ( 1 4 3 ) δ ( x 1 4 1 6 ) )
E y ( x , 1 12 ) = e i k n r L 12 e i π 4 3 E 0 ( x , 0 ) * ( e i π 3 δ ( x 1 2 ) + δ ( x 1 4 ) + e i π 3 δ ( x 5 12 ) )
E y ( x , 1 12 ) = e i k n r L 12 e i π 4 3 E 0 ( x , 0 ) * ( e + i π 3 δ ( x 1 12 ) + δ ( x 1 4 ) + e + i π 3 δ ( x 5 12 ) )
F ( x , 1 3 q ) = e i k n r L 3 q 3 q E 0 ( x ) * ( t = 0 t 3 q ( mod 2 ) 6 q 1 e i π 4 ( 1 t 2 3 q ) ( δ ( x 1 6 t 6 q ) δ ( x + 1 6 t 6 q ) ) )
F ( x + 1 3 q ) = e i k n r L 3 q 3 q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 6 q 1 ( e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 ( t + 2 q ) 2 3 q ) ) δ ( x 1 6 t 6 q ) )
e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 ( t + 2 q ) 2 3 q ) = e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 t 2 + 4 t q + 4 q 2 3 q ) = e i π 4 ( 1 t 2 3 q ) ( 1 e i π 3 ( t + q ) )
τ ( t ) = { 0 t 0 ( mod 3 ) 1 t 1 , 2 ( mod 3 ) 1 t 1 , 2 ( mod 3 )
e i π 4 ( 1 t 2 3 q ) = e i π 4 ( 1 ( 2 t q ) 2 3 q ) = e i π 4 ( 1 4 t 2 4 t q + q 2 3 q ) = e i π 4 e i π 3 q t 2 e i π 3 t e i π 12 q
e i π 4 ( 1 t 2 3 q ) ( 1 e i π 3 ( t + q ) ) = τ ( t ) i 3 e i π 3 t e i π 4 e i π 3 q t 2 e i π 3 t e i π 12 q = τ ( t ) 3 e i 3 π 4 e i π 3 q t 2 e i π 12 q
F ( x , 1 3 q ) = e i k n r L 3 q e i π ( 9 q 12 ) q E 0 ( x ) * ( t = 0 3 q 1 τ ( t ) e i π 3 q t 2 δ ( x t 3 q ) )
E y ( x , 1 3 q ) = e i k n r L 3 q e i π ( 9 q 12 ) q E 0 ( x ) * ( t = 0 3 q 2 τ ( t ) e i π 3 q t 2 δ ( x t 3 q ) )
F ( x , 1 3 q ) = e i k n r L 3 q 3 q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 6 q 1 e i π 4 ( 1 t 2 3 q ) ( δ ( x 2 6 t 6 q ) δ ( x + 2 6 t 6 q ) ) )
F ( x , 1 3 q ) = e i k n r L 3 q 3 q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 6 q 1 ( e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 ( t 2 q ) 2 3 q ) ) δ ( x 2 6 t 6 q ) )
e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 ( t 2 q ) 2 3 q ) = e i π 4 ( 1 t 2 3 q ) e i π 4 ( 1 t 2 4 q + 4 q 2 3 q ) = e i π 4 ( 1 t 2 3 q ) ( 1 e i π 3 ( q t ) )
F ( x , 1 3 q ) = e i k n r L 3 q 3 q E 0 ( x ) * ( t = 0 t q ( mod 2 ) 6 q 1 ( e i π 4 ( 1 t 2 3 q ) ( 1 e i π 3 ( q t ) ) ) δ ( x 2 6 t 6 q ) )
e i π 4 ( 1 t 2 3 q ) = e i π 4 ( 1 ( q 2 t ) 2 3 q ) = e i π 4 ( 1 ( 2 t q ) 2 3 q )
F ( x , 1 3 q ) = e i k n r L 3 q e i π ( 9 q 12 ) q E 0 ( x ) * ( t = 0 3 q 1 τ ( t ) e i π 3 q t 2 δ ( x 1 2 + t 3 q ) )
E y ( x , 1 3 q ) = e i k n r L 3 q e i π ( 9 q 12 ) q E 0 ( x ) * ( t = 0 3 q 2 τ ( t ) e i π 3 q t 2 δ ( x 1 2 + t 3 q ) )
1 2 a 3 q = b 3 q a + b = 3 q 2
E y ( x , z = 1 6 ) = e i k n r L 6 e i π ( 9 2 12 ) 2 E 0 ( x ) * ( A 1 ( τ ( 1 ) e i π 6 δ ( x 1 6 ) + τ ( 2 ) e i π 2 2 6 δ ( x 2 6 ) ) + A 2 ( τ ( 1 ) e i π 6 δ ( x 2 6 ) + τ ( 2 ) e i π 2 2 6 δ ( x 1 6 ) ) ) = e i k n r L 6 e i π ( 7 12 ) 2 E 0 ( x ) * ( ( A 1 e i π 6 + A 2 e i 2 π 3 ) δ ( x 1 6 ) + ( A 1 e i 2 π 3 + A 2 e i π 6 ) δ ( x 1 3 ) )
( B 1 B 2 ) = e i k n r L 6 e i π ( 5 12 ) 2 M ( A 1 A 2 )
M = ( 1 i i 1 )
( B 1 B 2 B 3 B 4 ) = e i k n r L 12 2 M ( A 1 A 2 )
M = ( e i π 3 e i 2 π 3 e i 3 π 4 e i 3 π 4 e i 3 π 4 e i 3 π 4 e i 2 π 4 e i π 3 )
E y ( x , 0 ) = s = 1 q E 0 ( x , 0 ) * ( A s ( 0 ) δ ( x + 1 4 q s 2 q ) )
E y ( x , z = p q ) = t = 1 q E 0 ( x , 0 ) * ( A t ( p ) δ ( x + 1 4 q t 2 q ) )
A ( p ) = e i k n r p L q ( p , q ) A ( 0 )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( s = 1 q t = 1 t q ( mod 2 ) 2 q e i π 4 ( 1 t 2 q ) A s ( 0 ) ( δ ( x + 1 4 q s 2 q t 2 q ) δ ( x 1 4 q + s 2 q t 2 q ) ) )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( s = 1 q t = 1 t s q ( mod 2 ) 2 q e i π 4 ( 1 ( t s ) 2 q ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) t = 1 t s q + 1 ( mod 2 ) 2 q e i π 4 ( 1 ( t + s 1 ) 2 q ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) ) = e i k n r L q E 0 ( x ) * ( s = 1 q t = 1 2 q α ( s , t ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) )
α ( s , t ) = { 1 2 e i π 4 ( 1 ( t s ) 2 q ) t s q ( mod 2 ) 1 q e i π 4 ( 1 ( t + s 1 ) 2 q ) t s q + 1 ( mod 2 )
σ ( t s , q ) = { 1 t s q ( mod 2 ) 0 t s q + 1 ( mod 2 )
α ( s , t ) = ( 1 ) σ q e i π 4 ( 1 ( t ( 1 ) σ s σ ) 2 q )
E y ( x , 0 ) = e i k n r L q E 0 ( x ) * ( s = 1 q t = 1 q α ( s , t ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) ) = E 0 ( x ) * ( t = 1 q A t ( 1 ) δ ( x + 1 4 q t 2 q ) )
( 1 , q ) s , t = α ( s , t ) = ( 1 ) σ q e i π 4 ( 1 ( t ( 1 ) σ σ ) 2 q )
˜ ( 1 , q ) s , t = α ( s , t ) = 1 q e i π 4 ( 1 ( t ( 1 ) σ σ ) 2 q )
( 1 , 4 ) = 1 2 ( e i π 4 1 1 e i π 4 1 e i π 4 e i π 4 1 1 e i π 4 e i π 4 1 e i π 4 1 1 e i π 4 )
E y ( x , 1 2 q ) e i k n r L 2 q E 0 ( x ) * ( s = 1 q t = 1 q ( 1 2 , q ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) ) = E 0 ( x ) * ( t = 1 q A t ( 1 2 ) δ ( x + 1 4 q t 2 q ) )
( 1 2 , q ) s , t = 2 q sin ( π 4 q ( 2 t 1 ) ( 2 s 1 ) ) e i π 4 ( 3 2 ( t 2 + s 2 t s ) 1 q )
˜ ( 1 2 , q ) s , t = 2 q cos ( π 4 q ( 2 t 1 ) ( 2 s 1 ) ) e i π 4 ( 1 2 ( t 2 + s 2 t s ) 1 q )
( 1 2 , 4 ) = 1 2 ( e i π 13 16 sin ( π 16 ) e i π 9 16 sin ( 3 π 16 ) e i π 16 sin ( 5 π 16 ) e i π 11 16 sin ( 7 π 16 ) e i π 9 16 sin ( 3 π 16 ) e i 5 π 16 sin ( 7 π 16 ) e 3 i π 16 sin ( π 16 ) e i π 16 sin ( 5 π 16 ) e i π 16 sin ( 5 π 16 ) e 3 i π 16 sin ( π 16 ) e i 5 π 16 sin ( 7 π 16 ) e i 9 π 16 sin ( 3 π 16 ) e 11 i π 16 sin ( 7 π 16 ) e i π 16 sin ( 5 π 16 ) e i 9 π 16 sin ( 3 π 16 ) e i 13 π 16 sin ( π 16 ) ) ( 0.14 e i π 13 16 0.4 e i π 9 16 0.16 e i π 16 0.7 e i π 11 16 0.4 e i π 9 16 0.7 e i 5 π 16 0.14 e 3 i π 16 0.6 e i π 16 0.6 e i π 16 0.14 e 3 i π 16 0.7 e i 5 π 16 0.4 e i 9 π 16 0.7 e i 3 π 16 0.6 e i π 16 0.4 e i 9 π 16 0.14 e i 13 π 16 )
E y ( x , 0 ) = E 0 ( x ) * ( s = 1 q 1 A s ( 0 ) δ ( x s 2 q ) )
E y ( x , z = p q ) = E 0 ( x ) * ( t = 1 q 1 A t ( p ) δ ( x t 2 q ) )
A ( p ) = e i k n r p L q 𝒩 ( p , q ) A ( 0 )
𝒩 ( 1 , q ) s , t = ( 1 σ ( s , t ) ) 2 q sin ( π t s 2 q ) e i π 4 ( 3 t 2 + s 2 q )
𝒩 ˜ ( 1 , q ) s , t = ( 1 σ ( s , t ) ) 2 q cos ( π t s 2 q ) e i π 4 ( 1 t 2 + s 2 q )
𝒩 ( 1 2 , q ) s , t = 2 q sin ( π t s q ) e i π 4 ( 3 2 ( t 2 + s 2 ) q )
𝒩 ˜ ( 1 2 , q ) s , t = 2 q cos ( π t s q ) e i π 4 ( 1 2 ( t 2 + s 2 ) q )
𝒩 ( 1 , 5 ) = 2 5 ( 0 e i π 2 sin ( π 5 ) 0 e i π 10 sin ( 2 π 5 ) e i π 2 sin ( π 5 ) 0 e i π 10 sin ( 2 π 5 ) 0 0 e i π 10 sin ( 2 π 5 ) 0 e i π 2 sin ( π 5 ) e i π 10 sin ( 2 π 5 ) 0 e i π 2 sin ( π 5 ) 0 ) ( 0 0.52 e i π 2 0 0.85 e i π 10 0.52 e i π 2 0 0.85 e i π 10 0 0 0.85 e i π 10 0 0.52 e i π 2 0.85 e i π 10 0 0.52 e i π 2 0 )
𝒩 ( 1 2 , 5 ) = 2 10 ( sin ( π 5 ) e i 11 π 20 sin ( 2 π 5 ) e i π 4 sin ( 2 π 5 ) e i π 4 sin ( π 5 ) e i 19 π 20 sin ( 2 π 5 ) e i π 4 sin ( π 5 ) e i π 20 sin ( π 5 ) e i 9 π 20 sin ( 2 π 5 ) e i π 4 sin ( 2 π 5 ) e i π 4 sin ( π 5 ) e i 9 π 20 sin ( π 5 ) e i π 20 sin ( 2 π 5 ) e i π 4 sin ( π 5 ) e i 19 π 20 sin ( 2 π 5 ) e i π 4 sin ( 2 π 5 ) e i π 4 sin ( π 5 ) e i 11 π 20 ) ( 0.37 e i 11 π 20 0.60 e i π 4 0.60 e i π 4 0.37 e i 19 π 20 0.60 e i π 4 0.37 e i π 20 0.37 e i 9 π 20 0.60 e i π 4 0.60 e i π 4 0.37 e i 9 π 20 0.37 e i π 20 0.60 e i π 4 0.37 e i 19 π 20 0.60 e i π 4 0.60 e i π 4 0.37 e i 11 π 20 )
Π ( x ) = { 1 | x | < 1 2 1 2 | x | = 1 2 0 | x | > 1 2
E y ( x , z ) s = 1 q E y ( s 2 q 1 4 q , z ) Π ( 2 q ( x + 1 4 q s 2 q ) ) = s = 1 q Π ( 2 q x ) * ( E y ( s 2 q 1 4 q , z ) δ ( x + 1 4 q s 2 q ) )
E y ( z + p q ) = e i k n r p L q ( p , q ) E y ( 0 ) = ( e i k n r L q ( 1 , q ) ) p E y ( 0 )
ϑ ( x , p q ) = m = e i π m 2 p q + 2 π i m x = m = e i 2 π m 2 p 2 q + 2 π i m x
ϑ ( x , p q ) = m = e i π m 2 p q + 2 π i m x = r = 0 2 q 1 n = e i 2 π ( n 2 q + r ) 2 p 2 q + 2 π i ( n 2 q + r ) x = r = 0 2 q 1 n = e i 2 π ( n 2 4 q 2 + 4 n q r + r 2 ) p 2 q + 4 π i n q x + 2 π i r x = r = 0 2 q 1 n = e 2 π i r 2 p 2 q e 4 π i n q x e 2 π i r x = r = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r x n = e 2 π i n ( 2 q x )
n = e 2 π i n ( 2 q x ) = n = δ ( 2 q x n ) = 1 2 q n = δ ( x n 2 q )
1 2 q r = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r x
ϑ ( x , p q ) = 1 2 q r = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r x t = 0 2 q 1 k = δ ( x ( 2 q ) k + t 2 q ) = 1 2 q r = 0 2 q 1 t = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r t 2 q k = δ ( x t 2 q k ) = 1 2 q r = 0 2 q 1 t = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r t 2 q δ ( x t 2 q )
1 2 q r = 0 2 q 1 e 2 π i r 2 p 2 q e 2 π i r t 2 q
1 b r = 0 b 1 e 2 π i ( a r 2 + n r b ) = c n ( a , b )
ϑ ( x , p q ) = t = 0 2 q 1 c t ( p , 2 q ) δ ( x t 2 q )
r = 0 | b | 1 e π i ( a r 2 + n r b ) = | b a | e π i ( | a b | n 2 4 a b ) r = 0 | a | 1 e π i ( b r 2 + n r a )
c t ( 1 , 2 q ) = 1 2 q q e π i ( q t 2 4 q ) r = 0 1 e π i ( 2 q r 2 + 2 t r 2 ) = 1 2 q e π i ( q t 2 4 q ) ( 1 + e π i ( q + t ) )
c t ( 1 , 2 q ) = 1 q e π i ( q t 2 4 q ) = 1 q e i π 4 e i π t 2 4 q
ϑ ( x , 1 q ) = t = 0 2 q 1 c t ( 1 , 2 q ) δ ( x t 2 q ) = 1 q t = 0 t q ( mod 2 ) 2 q 1 e i π 4 ( 1 t 2 q ) δ ( x t 2 q )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q t = 1 t 2 q ( mod 2 ) 4 q e i π 4 ( 1 t 2 2 q ) A s ( 0 ) ( δ ( x + 1 4 q s 2 q t 4 q ) δ ( x 1 4 q + s 2 q t 4 q ) ) )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q t = 1 2 q e i π 4 ( 1 2 t 2 q ) A s ( 0 ) ( δ ( x + 1 4 q s 2 q t 2 q ) δ ( x 1 4 q + s 2 q t 2 q ) ) )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q t = 1 2 q ( e i π 4 ( 1 2 ( t s ) 2 q ) e i π 4 ( 1 2 ( t + s 1 ) 2 q ) ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) )
β ( s , t ) = 1 2 q e i π 4 ( 1 2 ( t s ) 2 q ) e i π 4 ( 1 2 ( t + s 1 ) 2 q )
β ( s , t ) = 2 i 2 q sin ( π 8 2 q ( ( t + s 1 ) 2 ( t s ) 2 ) ) e i π 8 ( 1 2 ( t s ) 2 q + 1 2 ( t + s 1 ) 2 q ) = 2 q sin ( π 4 q ( 2 t 1 ) ( 2 s 1 ) ) e i π 4 ( 3 ( t s ) 2 + ( t + s 1 ) 2 q ) = 2 q sin ( π 4 q ( 2 t 1 ) ( 2 s 1 ) ) e i π 4 ( 3 2 ( t 2 + s 2 t s + 1 ) q )
E y ( x , 1 2 q ) = e i k n r L 2 q E 0 ( x ) * ( s = 1 q t = 1 q ( 1 2 , q ) A s ( 0 ) δ ( x + 1 4 q t 2 q ) )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( s = 1 q 1 t = 1 t q ( mod 2 ) 2 q e i π 4 ( 1 t 2 q ) A s ( 0 ) ( δ ( x s 2 q t 2 q ) δ ( x + s 2 q t 2 q ) ) )
F ( x , 1 q ) = e i k n r L q q E 0 ( x ) * ( s = 1 q 1 t = 1 t s q ( mod 2 ) 2 q ( e i π 4 ( 1 ( t s ) 2 q ) e i π 4 ( 1 ( t + s ) 2 q ) ) A s ( 0 ) δ ( x t 2 q ) ) = e i k n r L q E 0 ( x ) * ( s = 1 q 1 t = 1 2 q γ ( s , t ) A s ( 0 ) δ ( x t 2 q ) )
γ ( s , t ) = { 1 q e i π 4 ( 1 ( t s ) 2 q ) e i π 4 ( 1 ( t + s ) 2 q ) t s q ( mod 2 ) 0 t s q + 1 ( mod 2 )
γ ( s , t ) = ( 1 σ ( s , t ) ) q e i π 4 ( 1 ( t s ) 2 q ) e i π 4 ( 1 ( t + s ) 2 q )
e i π 4 ( 1 ( t s ) 2 q ) e i π 4 ( 1 ( t + s ) 2 q ) = 2 i sin ( π 8 q ( ( t + s ) 2 ( t s ) 2 ) ) e i π 8 ( 2 ( t s ) 2 + ( t + s ) 2 q ) = 2 sin ( π t s 2 q ) e i π 4 ( 3 t 2 + s 2 q )
E y ( x 1 q ) = e i k n r L q E 0 ( x ) * ( s = 1 q 1 t = 1 q 1 𝒩 ( 1 , q ) A s ( 0 ) δ ( x t 2 q ) )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q 1 t = 1 t 2 q ( mod 2 ) 4 q e i π 4 ( 1 t 2 2 q ) A s ( 0 ) ( δ ( x s 2 q t 4 q ) δ ( x + s 2 q t 4 q ) ) )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q 1 t = 1 2 q e i π 4 ( 1 2 t 2 q ) A s ( 0 ) ( δ ( x s 2 q t 2 q ) δ ( x + s 2 q t 2 q ) ) )
F ( x , 1 2 q ) = e i k n r L 2 q 2 q E 0 ( x ) * ( s = 1 q 1 t = 1 2 q ( e i π 4 ( 1 2 ( t s ) 2 q ) e i π 4 ( 1 2 ( t + s ) 2 q ) ) A s ( 0 ) δ ( x t 2 q ) ) = e i k n r L 2 q E 0 ( x ) * ( s = 1 q 1 t = 1 2 q ε ( s , t ) A s ( 0 ) δ ( x t 2 q ) )
ε ( s , t ) = 1 2 q e i π 4 ( 1 2 ( t s ) 2 q ) e i π 4 ( 1 2 ( t + s ) 2 q )
e i π 4 ( 1 2 ( t s ) 2 q ) e i π 4 ( 1 2 ( t + s ) 2 q ) = 2 i sin ( π 8 q 2 ( ( t + s ) 2 ( t s ) 2 ) ) e i π 8 ( 2 2 ( t s ) 2 + ( t + s ) 2 q ) = 2 sin ( π t s q ) e i π 4 ( 3 2 ( t 2 + s 2 ) q )
E y ( x 1 2 q ) = e i k n r L 2 q E 0 ( x ) * ( s = 1 q 1 t = 1 q 1 𝒩 ( 1 2 , q ) A s ( 0 ) δ ( x t 2 q ) )

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