1 N 1 d 2 Θ j A j ] exp [ i k ( ϵ j = 1 N 1 r j r j + 1 2 | Θ j + 1 Θ j ϵ | 2 m = 1 n ψ ( Θ l m ) ) ] , (24)

where the normalization is chosen

A j = 2 π i ϵ k r j r j + 1 , (25)

so that F = 1 in the limit ψ = 0 . Then, Equation (24) is rephrased as

F = [ j = 1 l 1 1 d 2 Θ j A j ] exp [ i k ( ϵ j = 1 l 1 1 r j r j + 1 2 | Θ j + 1 Θ j ϵ | 2 ) ] × [ j = l 1 l 2 1 d 2 Θ j A j ] exp [ i k ( ϵ j = l 1 l 2 1 r j r j + 1 2 | Θ j + 1 Θ j ϵ | 2 ψ ( Θ l 1 ) ) ] × × [ j = l n N 1 d 2 Θ j A j ] exp [ i k ( ϵ j = l n N 1 r j r j + 1 2 | Θ j + 1 Θ j ϵ | 2 ψ ( Θ l n ) ) ] , (26)

where ϵ is the separation between two neighboring planes. With the use of the following equality, which can be proven by the mathematical induction,

j = l m l m + 1 1 r j r j + 1 | Θ j + 1 Θ j | 2 = ϵ r l m r l m + 1 r l m + 1 r l m | Θ l m + 1 Θ l m | 2 + j = l m + 1 l m + 1 1 r j 2 r j + 1 r l m r j r l m | Θ j u l m , j | 2

with

u l m , j = r l m Θ l m + ( j l m ) r j + 1 Θ j + 1 j ( r j + 1 r l m ) , (27)

we have

#Math_98# (28)

which is equivalent to Eqution (7).

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact ijaa@scirp.org

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Takahashi, R. and Nakamura, T. (2003) Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries. The Astrophysical Journal, 595, 1039-1051.
https://doi.org/10.1086/377430
[2] Seto, N. (2004) Strong Gravitational Lensing and Localization of Merging Massive Black Hole Binaries with LISA. Physical Review D, 69, Article ID: 022002
https://doi.org/10.1103/PhysRevD.69.022002
[3] Yamamoto, K. and Tsunoda, K. (2003) Wave Effect in Gravitational Lensing by a Cosmic String. Physical Review D, 68, Article ID: 041302.
https://doi.org/10.1103/PhysRevD.68.041302
[4] De Paolis, F., Ingrosso, G., Nucita, A.A. and Qadir, A. (2002) A Note on Gravitational Wave Lensing. Astronomy and Astrophysics, 394, 749-752.
https://doi.org/10.1051/0004-6361:20021258
[5] Baraldo, C., Hosoya, A. and Nakamura, T.T. (1999) Gravitationally Induced Interference of Gravitational Waves by a Rotating Massive Object. Physical Review D, 59, Article ID: 083001.
https://doi.org/10.1103/PhysRevD.59.083001
[6] Nakamura, T.T. and Deguchi, S. (1999) Wave Optics in Gravitational Lensing. Progress of Theoretical Physics Supplement, 133, 137-153.
https://doi.org/10.1143/PTPS.133.137
[7] Nakamura, T.T. (1998) Gravitational Lensing of Gravitational Waves from Inspiraling Binaries by a Point Mass Lens. Physical Review Letters, 80, 1138-1141.
https://doi.org/10.1103/PhysRevLett.80.1138
[8] Deguchi, S. and Watson, W.D. (1986) Wave Effects in Gravitational Lensing of Electromagnetic Radiation. Physical Review D (Particles and Fields), 34, 1708-1718.
https://doi.org/10.1103/PhysRevD.34.1708
[9] Deguchi, S. and Watson, W.D. (1986) Diffraction in Gravitational Lensing for Compact Objects of Low Mass. Astrophysical Journal, Part 1, 307, 30-37.
https://doi.org/10.1086/164389
[10] Schneider, P., Ehlers, J. and Falco, E.E. (1992) Gravitational Lenses. Springer-Verlag, Berlin.
[11] Magnus, W., Oberhettinger, F. and Soni, R.P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, Berlin.
https://doi.org/10.1007/978-3-662-11761-3

  
comments powered by Disqus
IJAA Subscription
E-Mail Alert
IJAA Most popular papers
Publication Ethics & OA Statement
IJAA News
Frequently Asked Questions
Recommend to Peers
Recommend to Library
Contact Us

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.