Abstract

The paper is concerned with the problem of designing a planar array with nearly ideal radiation characteristics. An array of tapered waveguides is used, and each input Bloch mode is adiabatically transformed into a plane wave. The array then radiates most of its power in the Brillouin zone Ω of order m = 0. It can be useful for a variety of applications, including star couplers and wavelength multiplexers suitable for lightwave distribution in local area networks. A design procedure for optimizing the array is described, and optimum design conditions and optimum efficiency are explicitly derived.

© 1990 Optical Society of America

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References

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  1. C. Dragone, “Efficiency of a periodic array with nearly ideal element pattern,” IEEE Photon. Technol. Lett. 1, 238–249 (1989).
    [CrossRef]
  2. N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of. Phased Array Antennas (Wiley, New York, 1972), p. 149.
  3. H. Bach, J. E. Hansen, “Uniformly spaced arrays,” in Antenna Theory, Part 1, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 5, pp. 138–203.
  4. C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
    [CrossRef]
  5. N. A. Begovich, “Frequency scanning,” in Microwave Scanning Antennas, R. C. Hansen, ed. (Academic, New York, 1966), Chap. 2.
  6. M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
    [CrossRef]
  7. H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
    [CrossRef]
  8. J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  10. C. Dragone, “Scattering at a junction of two waveguides with different surface impedances,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1327 (1984).
    [CrossRef]
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–131.
  12. C. Dragone, “Efficient N× Nstar couplers using Fourier optics,” IEEE J. Lightwave Technol. 7, 479–489 (1989); see also C. Dragone, Electron. Lett. 24, 942–943 (1988).
    [CrossRef]
  13. F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers, Vol. 6 of IEE Electromagnetic Waves Series (Peregrinus, London, 1979), pp. 282–299.
  14. A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
    [CrossRef]
  15. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966), Chap. 6.

1990 (1)

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

1989 (3)

C. Dragone, “Efficiency of a periodic array with nearly ideal element pattern,” IEEE Photon. Technol. Lett. 1, 238–249 (1989).
[CrossRef]

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

C. Dragone, “Efficient N× Nstar couplers using Fourier optics,” IEEE J. Lightwave Technol. 7, 479–489 (1989); see also C. Dragone, Electron. Lett. 24, 942–943 (1988).
[CrossRef]

1988 (1)

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

1984 (1)

C. Dragone, “Scattering at a junction of two waveguides with different surface impedances,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1327 (1984).
[CrossRef]

1977 (1)

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

1975 (1)

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Amitay, N.

N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of. Phased Array Antennas (Wiley, New York, 1972), p. 149.

Bach, H.

H. Bach, J. E. Hansen, “Uniformly spaced arrays,” in Antenna Theory, Part 1, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 5, pp. 138–203.

Begovich, N. A.

N. A. Begovich, “Frequency scanning,” in Microwave Scanning Antennas, R. C. Hansen, ed. (Academic, New York, 1966), Chap. 2.

Burns, W. K.

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966), Chap. 6.

Dragone, C.

C. Dragone, “Efficient N× Nstar couplers using Fourier optics,” IEEE J. Lightwave Technol. 7, 479–489 (1989); see also C. Dragone, Electron. Lett. 24, 942–943 (1988).
[CrossRef]

C. Dragone, “Efficiency of a periodic array with nearly ideal element pattern,” IEEE Photon. Technol. Lett. 1, 238–249 (1989).
[CrossRef]

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

C. Dragone, “Scattering at a junction of two waveguides with different surface impedances,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1327 (1984).
[CrossRef]

Galindo, V.

N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of. Phased Array Antennas (Wiley, New York, 1972), p. 149.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hansen, J. E.

H. Bach, J. E. Hansen, “Uniformly spaced arrays,” in Antenna Theory, Part 1, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 5, pp. 138–203.

Harvey, G. T.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Henry, C. H.

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

Kaminow, I. P.

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

Kato, K.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Kistler, R. C.

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

Linke, R. A.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Lipson, J.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–131.

Milton, A. F.

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

Minford, W. J.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Murphy, E. J.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Nishi, I.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Rice, T. C.

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

Smit, M. K.

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

Sporleder, F.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers, Vol. 6 of IEE Electromagnetic Waves Series (Peregrinus, London, 1979), pp. 282–299.

Suzuki, S.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Takahashi, H.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Unger, H. G.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers, Vol. 6 of IEE Electromagnetic Waves Series (Peregrinus, London, 1979), pp. 282–299.

Wu, C. P.

N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of. Phased Array Antennas (Wiley, New York, 1972), p. 149.

Electron. Lett. (2)

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-wavelength grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

IEEE J. Lightwave Technol. (2)

J. Lipson, W. J. Minford, E. J. Murphy, T. C. Rice, R. A. Linke, G. T. Harvey, “A six-channel wavelength multiplexer and demultiplexer for single mode systems,” IEEE J. Lightwave Technol. LT-3, 1159–1163 (1975).

C. Dragone, “Efficient N× Nstar couplers using Fourier optics,” IEEE J. Lightwave Technol. 7, 479–489 (1989); see also C. Dragone, Electron. Lett. 24, 942–943 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

C. Dragone, “Efficiency of a periodic array with nearly ideal element pattern,” IEEE Photon. Technol. Lett. 1, 238–249 (1989).
[CrossRef]

C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. Dragone, “Scattering at a junction of two waveguides with different surface impedances,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1327 (1984).
[CrossRef]

Other (7)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–131.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers, Vol. 6 of IEE Electromagnetic Waves Series (Peregrinus, London, 1979), pp. 282–299.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966), Chap. 6.

N. A. Begovich, “Frequency scanning,” in Microwave Scanning Antennas, R. C. Hansen, ed. (Academic, New York, 1966), Chap. 2.

N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of. Phased Array Antennas (Wiley, New York, 1972), p. 149.

H. Bach, J. E. Hansen, “Uniformly spaced arrays,” in Antenna Theory, Part 1, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 5, pp. 138–203.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (11)

Fig. 1
Fig. 1

Width 2γ of the Brillouin zone Ω is determined by the period a of the periodic array. The power radiated in Ω is maximized by minimizing mode conversion in the transition.

Fig. 2
Fig. 2

Patterns η(θ) for two transitions optimized for L/a = 30βoa and L/a = 150βoa.

Fig. 3
Fig. 3

Plane-wave incident in the direction of θo is transformed by the arrangement (a) into a plane wave whose direction θ is a function of λ. The angular dispersion is specified according to Eq. (9) by the difference in length l = lili−1. It is possible, by using the array (a) and a suitable imaging arrangement, as in Ref. 8, to construct the frequency-selective coupler (b).

Fig. 4
Fig. 4

Function ψ(X), specifying t(υ) in a linear taper.

Fig. 5
Fig. 5

Transition profile V(z) for an optimized array.

Fig. 6
Fig. 6

Optimum design parameters and optimum performance m.

Fig. 7
Fig. 7

Optimized patterns calculated using Eq. (92) and relation (93).

Fig. 8
Fig. 8

Behavior, for σ = σc, of the coupling coefficient t and the differential phase υ for the case L/a = 150β0a.

Fig. 9
Fig. 9

Patterns of |Ψ(x, z)|2 for z = 0.58 zo and z = 0.

Fig. 10
Fig. 10

Patterns |Ψ(x, o)|2 for L/a = 30βoa.

Fig. 11
Fig. 11

Patterns η(θ) for L/a = 30β0a and L/a = 50β0a.

Equations (133)

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c s = exp ( j s ϕ ) , | ϕ | < π ,
| ϕ | < π
k a sin θ = ϕ ,
c s = δ s , r = 1 2 π π π exp [ j ( s r ) ϕ ] d ϕ ,
exp ( j r ϕ ) d ϕ ,
P ( θ ) = cos θ P ( 0 ) η ( 0 ) η ( θ ) ,
η ( ϕ ) 1 ,
P ( θ ) cos θ P ( 0 )
| θ | < γ ,
a sin γ = λ / 2 .
L / a = 150 k a
ϕ 0 = k a sin θ 0
ϕ = k a sin θ = k a sin θ 0 + 2 π l λ + 2 π q ,
( s + r p ) = N q ,
1 p N .
c s f ( x s a ) exp ( j β o z ) ,
ψ = s c s Ψ ( x s a , z ) ,
ψ = exp ( j σ x ) u ( x , z ) ,
σ = ϕ / a = k sin θ
Ψ ( x , z ) = 1 g C ( ξ , z ) exp ( j ξ x ) d ξ
u ( x , z ) = m C ( σ + m g , z ) exp ( j m g x ) ,
C ( ξ , z ) = A ( ξ ) exp [ j ( k 2 ξ 2 ) 1 / 2 z ] ,
Ψ a cos θ A ( k sin θ ) exp j k r ( j λ r ) 1 / 2 for r .
P ( θ ) = ( a 2 / λ ) | A ( k sin θ ) | 2 cos 2 θ .
ψ = m A m exp [ j k ( sin θ m x + cos θ m z ) ] ,
A m = A ( k sin θ m )
k sin θ m = σ + m g .
| a sin θ m λ | < λ / 2 ,
| θ 0 | 2 < | θ 1 | 2 < | θ 1 | 2 < .
( R e ) a / 2 a / 2 1 j k ψ z ψ * d z
P ι = a m | A m | 2 cos θ m .
η ( θ m ) = | A m | 2 cos θ m s | A s | 2 cos θ s .
P ( θ m ) cos θ m = D | A m | 2 cos θ m s | A s | 2 cos θ s = D η ( θ m ) ,
D = P t a λ .
D = m P ( θ m ) cos θ m ,
1 η ( θ o ) = m 0 η ( θ m )
ψ m A m exp [ j ( σ + m g ) x + j β m z ] for z 0.
z j β m ,
β o β 1 ,
0 σ a < π .
β o β m , for m 0 ,
( 2 x 2 + 2 z 2 + k 2 ) ψ = 0 ,
( k a ) 2 = V o + s 0 V s exp ( j s g x )
ψ = u exp [ j σ x + j Φ ( z ) ] ,
u = r A r exp ( j r g x ) ,
Φ = z β d z ,
exp [ j ( σ + s g ) x ]
[ V o ( β a ) 2 ( σ a + 2 s π ) 2 ] A s + r V r A s r = 0 ( s = 0 , ± 1 , ) .
V s 0 for s 0.
| A s | | A m | for s m .
β m [ k 2 ( σ + m g ) 2 ] 1 / 2 .
β 0 β 1 .
A s V s ( σ a + 2 s π ) 2 V o + ( β a ) 2 A o ,
( β a ) 2 V o ( σ a ) 2 s V s V s V o ( β a ) 2 ( σ a 2 s π ) 2 ,
σ a π .
( β o a ) 2 V o ( σ a ) 2 2 π ( π σ a ) + [ 4 π 2 ( π σ a ) 2 + V 2 ] 1 / 2 ,
V 2 = V 1 V 1 .
( β 1 a ) 2 V o ( σ a ) 2 2 π ( π σ a ) [ 4 π 2 ( π σ a ) 2 + V 2 ] 1 / 2 .
β o 2 β 1 2 for σ a π .
β 0 > β 1 β 1 >
0 σ a < π .
β o β 1
( ψ m , ψ n ) = ( u m , u n ) = 0 for m n ,
( u m , u n ) = 1 a a / 2 a / 2 u m u n * d x .
1 2 j k [ ( ψ z , ψ ) ( ψ , ψ z ) ] = β k ( u , u ) ,
( u , u ) = k β .
u 0 = ( k β 0 ) 1 / 2 1 ( 1 + | α | 2 ) 1 / 2 [ 1 + α exp ( j 2 π a x ) ]
u 1 = ( k β 1 ) 1 / 2 1 ( 1 + | α | 2 ) 1 / 2 [ α * + exp ( j 2 π a x ) ] ,
α = μ 1 + ( 1 + | μ | 2 ) 1 / 2
μ = V 1 2 π δ σ a ,
δ σ a = π σ a .
( u o , u m z ) δ z = ( u o z , u m ) δ z ,
A u o ( x ) exp [ j Φ 0 ( z ) ] .
A T m u m ( x ) exp [ j Φ 0 , m ( z ) ] ,
Φ 0 , m = ξ z ( β o β m ) d z ( z > ξ )
A R m u m ( x ) exp [ j Φ 0 , m ( z ) ] ,
Φ 0 , m ( z ) = z ξ ( β o + β m ) d z ( ξ > z ) ,
( 1 + R o ) u o = T o u o + s 0 ( T s R s ) u s + δ a ,
( 1 R o ) β o u o = β o T o u o + s 0 β s ( T s + R s ) u s + δ b ,
δ a = s 0 R s ( u s u s ) , δ b = s 0 R s ( β s u s β s u s ) ,
R o = β o β 0 β o + β 0
T m = ( 1 R 0 ) β 0 + ( 1 + R 0 ) β m 2 β m ( u o , u m ) ( u m , u m ) ,
R m = ( 1 R 0 ) β 0 ( 1 + R 0 ) β m 2 β m ( u o , u m ) ( u m , u m ) .
δ u m = u m u m , δ β s = β s β s
β m β o β o 1 ,
T m ( u o , u m ) ( u m , u m ) .
( u o , u m ) ( u o , u m z ) δ z ,
δ T m T m z δ z ,
T m z ( u o , u m z ) β o + β m 2 β m .
A m exp ( j Φ 0 ) ( cos θ m ) 1 / 2 z o 0 A T z exp ( j υ m ) d z ,
υ m = z 0 [ β o ( z ) β m ( z ) ] d z ( z 0 ) .
A o A exp [ j Φ o ( 0 ) ] ,
A m A o z o 0 T m z exp ( j υ m ) d z .
T m z R m z , β m β o β m + β o .
A m A o [ T m + z o 0 T m z exp ( j υ m ) d z ] ,
T 1 α o ( 1 + | α o | 2 ) 1 / 2 ,
T 1 z 1 1 + | α | 2 α z .
| θ | < γ c ,
= η c 2 sin γ c sin γ .
τ = t exp ( j υ ) d υ ,
V ( z ) = 2 π B p ( 1 p 2 ) 1 / 2 ,
p = 3 2 z z o [ 1 1 3 ( z z o ) 2 ] .
y = z z o = z L ,
τ = 0 1 d T d y exp ( j υ ) d y ,
w = w ( y ) = d υ d y = L ( β o β m )
d T d y = 1 2 ( 1 + μ 2 ) d μ d y
υ = w ( 0 ) 0 y ( 1 + μ 2 ) 1 / 2 d y , w ( 0 ) = L a 2 π δ σ a β 0 a ,
δ σ a = π σ a , μ = V 2 π δ σ a .
t = d T d υ = 1 W 1 2 ( 1 + μ 2 ) 3 / 2 , W = L a ( 2 π δ σ a ) 2 β o a 1 d V / d y .
X = 1 2 ( sinh 1 μ + μ ( 1 + μ 2 ) , Y = 1 2 ( 1 + μ 2 ) 3 / 2 .
τ = 0 Y ( X ) exp ( j W X ) d X ,
τ j 1 2 W ,
t = c ( 1 y m ) ,
2 w ( 0 ) t d y = d μ ( 1 + μ 2 ) 3 / 2 ,
μ = p ( y ) [ 1 p 2 ( y ) ] 1 / 2 , p ( y ) = 2 c w ( 0 ) [ y y m + 1 m + 1 ] .
B = π σ c a .
p ( y ) = 3 2 ( y 1 3 y 3 ) ,
y = ( 1 y o ) z z o + y o .
τ = [ μ 2 ( 1 + μ 2 + 1 + μ 2 ) 1 / 2 ] y o + y o 1 d T d y exp ( j υ ) d y ,
η 1 1 + | τ | 2 .
L a 1 β 0 a L a 1 k a
σ c a , B = π σ c a , y o ,
L a 1 β 0 a = 30 , 50 , 150 ,
L a 1 β 0 a > 150 ,
V 2 π n Δ n k 0 2 a 2 sin ( l π a ) ,
a = 2 l , β 0 a = 60
V ( z o ) = 60 ,
L a = 150 β o a
z z 0 = 0.58 , 0.
L a = 30 β 0 a ,
η ( θ 0 ) 1 η ( θ 1 ) .
0.69 , 0.78 , 0.90 ,
0.67 , 0.77 , 0.90 ,

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