Abstract

Free-space optical interconnects (FSOIs), made up of dense arrays of vertical-cavity surface-emitting lasers, photodetectors and microlenses can be used for implementing high-speed and high-density communication links, and hence replace the inferior electrical interconnects. A major concern in the design of FSOIs is minimization of the optical channel cross talk arising from laser beam diffraction. In this article we introduce modifications to the mode expansion method of Tanaka et al. [IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972)] to make it an efficient tool for modelling and design of FSOIs in the presence of diffraction. We demonstrate that our modified mode expansion method has accuracy similar to the exact solution of the Huygens-Kirchhoff diffraction integral in cases of both weak and strong beam clipping, and that it is much more accurate than the existing approximations. The strength of the method is twofold: first, it is applicable in the region of pronounced diffraction (strong beam clipping) where all other approximations fail and, second, unlike the exact-solution method, it can be efficiently used for modelling diffraction on multiple apertures. These features make the mode expansion method useful for design and optimization of free-space architectures containing multiple optical elements inclusive of optical interconnects and optical clock distribution systems.

© 2003 Optical Society of America

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2002 (2)

2000 (6)

X. Zheng, P. J. Marchand, D. Huang, O. Kibar, S. C. Esener, “Cross talk and ghost talk in a microbeam free-space optical interconnect system with vertical-cavity surface-emitting lasers, microlenses, and metal-semiconductor-metal detectors,” Appl. Opt. 39, 4834–4841 (2000).
[CrossRef]

D. F.- Brosseau, F. Lacroix, M. H. Ayliffe, E. Bernier, B. Robertson, F. A. P. Tooley, D. V. Plant, A. G. Kirk, “Design, implementation, and characterization of a kinematically aligned, cascaded spot-array generator for a modulator-based free-space optical interconnect,” Appl. Opt. 39, 733–745 (2000).
[CrossRef]

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
[CrossRef]

D. V. Plant, A. G. Kirk, “Optical interconnects at the chip and board level: Challenges and solutions,” Proc. IEEE 88, 806–818 (2000).
[CrossRef]

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

N. McArdle, M. Naruse, H. Toyoda, Y. Kobayashi, M. Ishikawa, “Reconfigurable optical interconnections for parallel computing,” Proc. IEEE 88, 829–837 (2000).
[CrossRef]

1999 (1)

R. Wang, A. D. Rakić, M. L. Majewski, “Analysis of lensless free-space optical interconnects based on multi-transverse mode vertical-cavity-surface-emitting lasers,” Opt. Commun. 167, 261–271 (1999).
[CrossRef]

1998 (1)

1997 (3)

1996 (1)

C. J. Kuo, Y. S. Su, H. T. Chang, “Wavelength-division microlens interconnection using weakly diffracted Gaussian beam,” Opt. Quantum Electron. 28, 381–394 (1996).
[CrossRef]

1994 (1)

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

1993 (4)

D. H. Martin, J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[CrossRef]

J. A. Murphy, S. Withington, A. Egan, “Mode conversion at diffracting apertures in millimeter and submillimeter wave optical systems,” IEEE Trans. Microwave Theory Tech. 41, 1700–1702 (1993).
[CrossRef]

J. A. Murphy, A. Egan, S. Withington, “Truncation in millimeter and submillimeter-wave optical systems,” IEEE Trans. Antennas Propag. 41, 1408–1413 (1993).
[CrossRef]

J. A. Murphy, A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

1992 (3)

S. Withington, J. A. Murphy, “Analysis of diagonal horns through Gaussian-Hermite modes,” IEEE Trans. Antennas Propag. 40, 198–206 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

1987 (1)

C. Campbell, “Fresnel diffraction of Gaussian laser beams by circular apertures,” Opt. Eng. 26, 270–275 (1987).
[CrossRef]

1982 (1)

1976 (1)

1972 (1)

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

1971 (1)

1970 (1)

1969 (1)

1966 (1)

Andrade, O. O.

Ayliffe, M. H.

Bartelt, H.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Belland, P.

Bernier, E.

Boisset, G. C.

Bowen, J. W.

D. H. Martin, J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[CrossRef]

Brosseau, D. F.-

Campbell, C.

C. Campbell, “Fresnel diffraction of Gaussian laser beams by circular apertures,” Opt. Eng. 26, 270–275 (1987).
[CrossRef]

Campbell, J. P.

Chang, H. T.

C. J. Kuo, Y. S. Su, H. T. Chang, “Wavelength-division microlens interconnection using weakly diffracted Gaussian beam,” Opt. Quantum Electron. 28, 381–394 (1996).
[CrossRef]

Chen, R. T.

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

Cloonan, T. J.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

Crenn, J. P.

DeShazer, L. G.

Egan, A.

J. A. Murphy, A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

J. A. Murphy, S. Withington, A. Egan, “Mode conversion at diffracting apertures in millimeter and submillimeter wave optical systems,” IEEE Trans. Microwave Theory Tech. 41, 1700–1702 (1993).
[CrossRef]

J. A. Murphy, A. Egan, S. Withington, “Truncation in millimeter and submillimeter-wave optical systems,” IEEE Trans. Antennas Propag. 41, 1408–1413 (1993).
[CrossRef]

Erhard, W.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Esener, S. C.

Fedor, A.

Feldblum, A. Y.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

Fey, D.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Fukumitsu, O.

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Garrett, L.

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

Gerold, D.

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Grimm, G.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Gruber, M.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Hinton, H. S.

Y. Liu, B. Robertson, D. V. Plant, H. S. Hinton, W. M. Robertson, “Design and characterization of a microchannel optical interconnect for optical backplanes,” Appl. Opt. 36, 3127–3141 (1997).
[CrossRef] [PubMed]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

Hoppe, L.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Huang, D.

Ishikawa, M.

N. McArdle, M. Naruse, H. Toyoda, Y. Kobayashi, M. Ishikawa, “Reconfigurable optical interconnections for parallel computing,” Proc. IEEE 88, 829–837 (2000).
[CrossRef]

Iyer, R.

Jahns, J.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Kibar, O.

Kirk, A. G.

Kobayashi, Y.

N. McArdle, M. Naruse, H. Toyoda, Y. Kobayashi, M. Ishikawa, “Reconfigurable optical interconnections for parallel computing,” Proc. IEEE 88, 829–837 (2000).
[CrossRef]

Kogelnik, H.

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1989).

Kuo, C. J.

C. J. Kuo, Y. S. Su, H. T. Chang, “Wavelength-division microlens interconnection using weakly diffracted Gaussian beam,” Opt. Quantum Electron. 28, 381–394 (1996).
[CrossRef]

Lacroix, F.

Li, G.

Li, M. M.

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

Li, T.

Liu, Y.

Majewski, M. L.

R. Wang, A. D. Rakić, M. L. Majewski, “Design of microchannel free-space optical interconnects based on vertical-cavity surface-emitting laser arrays,” Appl. Opt. 41, 3469–3478 (2002).
[CrossRef] [PubMed]

R. Wang, A. D. Rakić, M. L. Majewski, “Analysis of lensless free-space optical interconnects based on multi-transverse mode vertical-cavity-surface-emitting lasers,” Opt. Commun. 167, 261–271 (1999).
[CrossRef]

Marchand, P. J.

Martin, D. H.

D. H. Martin, J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[CrossRef]

McArdle, N.

N. McArdle, M. Naruse, H. Toyoda, Y. Kobayashi, M. Ishikawa, “Reconfigurable optical interconnections for parallel computing,” Proc. IEEE 88, 829–837 (2000).
[CrossRef]

McCormick, F. B.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

Mersereau, K. O.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
[CrossRef]

D. A. B. Miller, “Invited paper: Physical reasons for optical interconnection,” Int. J. Optoelectron. 11, 155–168 (1997).

Morozov, V.

Murphy, J. A.

J. A. Murphy, S. Withington, A. Egan, “Mode conversion at diffracting apertures in millimeter and submillimeter wave optical systems,” IEEE Trans. Microwave Theory Tech. 41, 1700–1702 (1993).
[CrossRef]

J. A. Murphy, A. Egan, S. Withington, “Truncation in millimeter and submillimeter-wave optical systems,” IEEE Trans. Antennas Propag. 41, 1408–1413 (1993).
[CrossRef]

J. A. Murphy, A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

S. Withington, J. A. Murphy, “Analysis of diagonal horns through Gaussian-Hermite modes,” IEEE Trans. Antennas Propag. 40, 198–206 (1992).
[CrossRef]

Naruse, M.

N. McArdle, M. Naruse, H. Toyoda, Y. Kobayashi, M. Ishikawa, “Reconfigurable optical interconnections for parallel computing,” Proc. IEEE 88, 829–837 (2000).
[CrossRef]

Neff, J.

Olaofe, G. O.

Ozguz, V. H.

Plant, D. V.

Rakic, A. D.

R. Wang, A. D. Rakić, M. L. Majewski, “Design of microchannel free-space optical interconnects based on vertical-cavity surface-emitting laser arrays,” Appl. Opt. 41, 3469–3478 (2002).
[CrossRef] [PubMed]

R. Wang, A. D. Rakić, M. L. Majewski, “Analysis of lensless free-space optical interconnects based on multi-transverse mode vertical-cavity-surface-emitting lasers,” Opt. Commun. 167, 261–271 (1999).
[CrossRef]

Robertson, B.

Robertson, W. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Sasian, J. M.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, S465–S477 (1992).
[CrossRef]

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, H. S. Hinton, K. O. Mersereau, A. Y. Feldblum, “Corrigendum: ‘Optical interconnections using microlens arrays’,” Opt. Quantum Electron. 24, 1209–1212 (1992).
[CrossRef]

Schell, R. G.

Shibukawa, M.

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1941).

Sinzinger, S.

D. Fey, W. Erhard, M. Gruber, J. Jahns, H. Bartelt, G. Grimm, L. Hoppe, S. Sinzinger, “Optical interconnects for neural and reconfigurable VLSI architectures,” Proc. IEEE 88, 838–848 (2000).
[CrossRef]

Su, Y. S.

C. J. Kuo, Y. S. Su, H. T. Chang, “Wavelength-division microlens interconnection using weakly diffracted Gaussian beam,” Opt. Quantum Electron. 28, 381–394 (1996).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Tang, S.

S. Tang, R. T. Chen, L. Garrett, D. Gerold, M. M. Li, “Design limitations of highly parallel free-space optical interconnects based on arrays of vertical cavity surface-emitting laser diodes, microlenses, and photodetectors,” J. Lightwave Technol. 12, 1971–1975 (1994).
[CrossRef]

Tooley, F. A. P.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a microchannel free-space optical interconnect.

Fig. 2
Fig. 2

Diagram for the general laser-beam diffraction problem.

Fig. 3
Fig. 3

(a) Schematic of the laser-beam diffraction problem in an FSOI: For a given set of design parameters the goal is to calculate the power received by each receiver. (b) The modified mode-expansion method enables us to simplify the above problem by replacing the original incident beam and diffracting aperture by the effective multi-modal beam given by Eqs. (11) and (22).

Fig. 4
Fig. 4

Approximating the diffraction field given by Eq. (7) (solid line) by the mode expansion method (large dots) (a) in the profile-matching sense, and (b) in the encircled power sense. Aperture A is assumed to be empty and d = 10.4 mm, κ = 1.6 (a = 120.3 μm).

Fig. 5
Fig. 5

Encircled power calculated directly from Eq. (7) and by the mode expansion method with different number of modes in the expanding beam.

Fig. 6
Fig. 6

Encircled power calculated by use of different methods on (a) receiver S, and (b) on receiver N: direct integration (solid curve), mode expansion method (large dots), the method of Belland and Crenn (small dots), and the method of Tang et al. (broken curve). Aperture A is assumed to be empty (distance from aperture A to the observation plane is d = 2.6 mm).

Fig. 7
Fig. 7

Encircled power calculated by use of different methods on (a) receiver S, and (b) on receiver N: direct integration (solid curve), mode expansion method (large dots), and the method of Belland and Crenn (small dots). Aperture A is assumed to contain a microlens with f = 800 μm (distance from aperture A to the observation plane is d = 10.4 mm).

Fig. 8
Fig. 8

Calculations of encircled power vs. receiver radius a s for 0 ≤ a s ≤ 125 μm and for three different clipping ratios calculated by: direct integration (solid curve), mode expansion method (large dots), and the method of Belland and Crenn (broken curve). Aperture A is empty and d = 10.4 mm.

Equations (34)

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Ur, θ, z=jk2πz-z0exp-jkz-z0×02π0aexp-jkr2+r02-2rr0 cosθ-θ02z-z0×ψr0, θ0, z0r0dr0dθ0,
ψnmr, θ, z=AnmzNnmr2wmLnm2r2w2×exp-r2w2-j kr22Rcosmθ,
Anmz=expj2n+m+1arctanλz+zsπws2-kz+zs
Nnm=2wπ1+δ0mn!n+m!1/2.
R=Rz=z1+πws2λz+zs2,
w=wz=ws1+λz+zsπws221/2.
Unmr, θ, z=2n!π1+δ0mn+m!×exp-jkz+zskjm+12η0z-z0cosmθexpj2n+m+1arctan ξ0-jkr22z-z0×kr2η0z-z0mp=0q=0n-1p+qp+q+m!p!q!p+m!×kr2η0z-z02pn+mn-q2τ2p+q+m+1×1-exp-12 η02τ2a2s=0p+q+m1s!η02τ2a22s.
ξ0=2z0+zskws2,
η0=2ws1+ξ02,
τ2=1+jξ0+jkη02z-z0.
Unmr, θ, z=n,m Cnmnmψnmr, θ, z.
Cnmnm=002π Unmr, θ, zψnm*r, θ, zrdrdθ,
Cnmnm=n!n+m!n!n+m!exp-jkzs-zs+j2n+m+1arctan ξ0-j2n+m+1arctan ξ0×p=0nq=0n-1p+qp+m+q!p!q!n+mn-q×n+mn-p2Bp+q+m+1η0η02p+m+1×1-exp-12 η02a2Bs=0p+q+m1s!η02a2B2s
Cnmnm=0
B=η02+η02η02+j ξ02η02-ξ02η02η02.
C00n0=exp-j2narctanξ0×p=0nnn-p-1qp! γ1+p, 2κ2,
κ=aw0
Cnmnm=002π Unmr, θ, zψnm*r, θ, zrdrdθ.
Cnmnm=002π Unmr0, θ0, z0ψnm*r0, θ0, z0×r0dr0dθ0,
Unmr0, θ0, z0=ψnmr0, θ0, z0
Unmr0, θ0, z0=0
Cnmnm=0a02π ψnmr0, θ0, z0ψnm*r0, θ0, z0×r0dr0dθ0,
ws=ws1-exp-a2ws21+2d1/kws22×coska22d11+kws2/2d12,
Is=Is1-2 exp-a2ws21+2d1/kws22×coska22d11+kws2/2d12-1.
Parx=Palens1-J0kalensarxd2-J1kalensarxd2,
Palens=1-J0kavcselalensd12-J1kavcselalensd12,
Cnmnm=0a02π ψnmr0, θ0, z0ψnm*r0, θ0, z0×r0dr0dθ0.
Cnmnm=2n!n+m!2n!n+m!exp-jkzs-zs×η0η0m+1 expj2n+m+1arctan ξ0-j2n+m+1arctan ξ0×0a r02m+1Lnmη02r02Lnmη02r02exp-σr02dr0,
σ=η02+η022+j η02ξ0-η02ξ02.
Cnmnm=2n!n+m!2n!n+m!exp-jkzs-zs×expj2n+m+1arctan ξ0-j2n+m+1arctan ξ0×p=0nq=0nn+mn-pn+mn-p-1p+qp!q!×η02p+m+1η02q+m+1σm+p+q+1 γm+p+q+1, a2σ
σ=η02B2,
γm+p+q+1, a2σ=m+p+q!×1-exp-12 η02a2Bs=0p+q+m1s!η02a2B2s,
Cnmnm=n!n+m!n!n+m!exp-jkzs-zs+j2n+m+1arctan ξ0+j2n+m+1arctan ξ0×p=0nq=0n-1p+qp+m+q!p!q!n+mn-q×n+mn-q2Bp+q+m+1η0η02p+m+1×1-exp-12 η02a2Bs=0p+q+m1s!η02a2B2s
Cnmnm=0

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