Abstract

We propose a design of a dielectric (silicon nitride) optical leaky wave antenna (OLWA) with periodic semiconductor (silicon) corrugations, capable of producing narrow beam radiation. The optical antenna radiates a narrow beam because a leaky wave (LW) with low attenuation constant is excited at one end of the corrugated dielectric waveguide. We show that pointing angle, beam-width, and operational frequency are all related to the LW complex wavenumber, whose value depends on the amount of silicon perturbations in the waveguide. In this paper, the propagation constant and the attenuation coefficient of the LW in the periodic structure are extracted from full-wave simulations. The far-field radiation patterns in both glass and air environments predicted by LW theory agree well with the ones obtained by full-wave simulations. We achieve a directive radiation pattern in glass environment with about 17.5 dB directivity and 1.05 degree beam-width at the operative free space wavelength of 1.55 μm, pointing at a direction orthogonal to the waveguide (broadside direction). We also show that the use of semiconductor corrugations facilitate electronic tuning of the radiation pattern via carrier injection.

© 2011 OSA

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2009 (6)

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009).
[CrossRef] [PubMed]

K. Van Acoleyen, W. Bogaerts, J. Jágerská, N. Le Thomas, R. Houdré, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009).
[CrossRef] [PubMed]

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

2006 (3)

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24(3), 1433–1439 (2006).
[CrossRef]

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[CrossRef]

2005 (1)

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

2002 (2)

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002).
[CrossRef]

1995 (1)

C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995).
[CrossRef]

1988 (1)

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988).
[CrossRef]

1981 (1)

W. P. Dumke, “Minority-carrier injection and storage into a heavily doped emitter—approximate solution for Auger recombination,” Solid-State Electron. 24(2), 155–157 (1981).
[CrossRef]

Akdas, M.

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Baets, R.

Barnes, W. L.

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

Bogaerts, W.

Boreman, G. D.

Boyraz, O.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Caglayan, H.

Cakmak, A. O.

Campbell, J. C.

S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002).
[CrossRef]

Capolino, F.

Cheben, P.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Chen, J.

Cherukulappurath, S.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008).
[CrossRef] [PubMed]

Colak, E.

Csutak, S. M.

S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002).
[CrossRef]

Dakshina-Murthy, S.

S. M. Csutak, S. Dakshina-Murthy, and J. C. Campbell, “CMOS-compatible planar silicon waveguide-grating-coupler photodetectors fabricated on silicon-on-insulator (SOI) substrates,” IEEE J. Quantum Electron. 38(5), 477–480 (2002).
[CrossRef]

Delage, A.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Della Villa, A.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “Minority-carrier injection and storage into a heavily doped emitter—approximate solution for Auger recombination,” Solid-State Electron. 24(2), 155–157 (1981).
[CrossRef]

Ebbesen, T. W.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Enoch, S.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Friedman, L.

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988).
[CrossRef]

Galdi, V.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Ghenuche, P.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008).
[CrossRef] [PubMed]

Gondarenko, A.

Houdré, R.

Jackson, D. R.

Jágerská, J.

Janz, S.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Jones, A. C.

Krenz, P. M.

Lamontagne, B.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Le Thomas, N.

Lewen, G. D.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Lezec, H. J.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Linke, R. A.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Lipson, M.

Lorenzo, J. P.

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988).
[CrossRef]

Manipatruni, S.

Manolatou, C.

Meyer, J.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

Micco, A.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Nahata, A.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Oliner, A. A.

Olmon, R. L.

Ozbay, E.

Pellerin, K. M.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Pierro, V.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Poitras, C. B.

Pradhan, S.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Preston, K.

Qian, F.

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

Qiang, R.

Quidant, R.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008).
[CrossRef] [PubMed]

Raschke, M. B.

Reed, G. T.

C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995).
[CrossRef]

Sage, I.

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

Sang, X.

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Sang, X. Z.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

Schmidt, B.

Shakya, J.

Song, Q.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[CrossRef]

Soref, R. A.

L. Friedman, R. A. Soref, and J. P. Lorenzo, “Silicon double-injection electro-optic modulator with junction gate control,” J. Appl. Phys. 63(6), 1831–1839 (1988).
[CrossRef]

Taminiau, T. H.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008).
[CrossRef] [PubMed]

Tanev, S.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Tang, C. K.

C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995).
[CrossRef]

Tayeb, G.

A. Micco, V. Galdi, F. Capolino, A. Della Villa, V. Pierro, S. Enoch, and G. Tayeb, “Directive emission from defect-free dodecagonal photonic quasicrystals: a leaky wave characterization,” Phys. Rev. B 79(7), 075110 (2009).
[CrossRef]

Thio, T.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002).
[CrossRef]

Tien, E. K.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

Tien, E.-K.

O. Boyraz, X. Sang, E.-K. Tien, Q. Song, F. Qian, and M. Akdas, “Silicon based optical pulse shaping and characterization,” Proc. SPIE 7212, 72120U (2009).
[CrossRef]

Tomov, I.

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

Van Acoleyen, K.

van Hulst, N. F.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett. 101(11), 116805 (2008).
[CrossRef] [PubMed]

Villa, A. D.

Wasey, J. A. E.

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

Wedge, S.

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

Xu, D. X.

P. Cheben, S. Janz, D. X. Xu, B. Lamontagne, A. Delage, and S. Tanev, “A broad-band waveguide grating coupler with a subwavelength grating mirror,” IEEE Photon. Technol. Lett. 18(1), 13–15 (2006).
[CrossRef]

Xu, Q.

Xu, Q. F.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

Q. Song, F. Qian, E. K. Tien, I. Tomov, J. Meyer, X. Z. Sang, and O. Boyraz, “Imaging by silicon on insulator waveguides,” Appl. Phys. Lett. 94(23), 231101 (2009).
[CrossRef]

S. Wedge, J. A. E. Wasey, W. L. Barnes, and I. Sage, “Coupled surface plasmon-polariton mediated photoluminescence from a top-emitting organic light-emitting structure,” Appl. Phys. Lett. 85(2), 182–184 (2004).
[CrossRef]

Electron. Lett. (1)

C. K. Tang and G. T. Reed, “Highly efficient optical-phase modulator in SOI waveguides,” Electron. Lett. 31(6), 451–452 (1995).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Optical antenna based on SOI. The Si periodic perturbation ( n S i = 3.48 ) is inside the Si3N4 waveguide ( n w = 1.67 ). The waveguide has a width equal to t w = 1 μm, is upon a SiO2 substrate ( n h = 1.45 ) and is covered by a SiO2 layer. The Si perturbation is characterized by an optimized period d = 970 nm, a width w = 300 nm and a length l = d / 2 = 485 nm, which are found after a parametric study. The number of elements of the perturbation is equal to 60. The red contour represents the boundary at which the field has been extracted for far-field calculations.

Fig. 2
Fig. 2

Comparison of the S-parameters obtained by COMSOL and HFSS with respect to the free space wavelength λ 0 varying from 1.4 µm to 1.7 µm.

Fig. 3
Fig. 3

Comparison between COMSOL and HFSS full-wave simulation results for the electric field along the waveguide, sampled in the silicon perturbations (one point per cell). (a) The magnitude and (b) the phase of the electric field along the aperture.

Fig. 4
Fig. 4

Comparison of the normalized far-field radiation patterns in free space, obtained by COMSOL (red line), HFSS (black line), EA method (green line) and AF method (blue line). The inset shows an enlargement of the maximum radiation region.

Fig. 5
Fig. 5

COMSOL results of the normalized far-field radiation pattern in free space by varying the free space wavelength λ 0 varying from 1.49 µm to1.56 µm.

Fig. 6
Fig. 6

Comparison of the normalized far-field radiation patterns in SiO2, obtained by COMSOL (red line), HFSS (black line), EA method (green line) and AF method (blue line). The inset shows an enlargement of the maximum radiation region.

Fig. 7
Fig. 7

COMSOL results of the normalized far-field radiation pattern in SiO2, by varying the free space wavelength λ 0 from 1.49 µm to1.56 µm.

Fig. 8
Fig. 8

The variation of (a) the silicon refractive index and (b) the normalized attenuation constant in silicon versus the injected carrier density. The carrier density N h = N e varies from 10 16 cm 3 to 10 19  cm 3 .

Fig. 9
Fig. 9

The variation of Qat broadside versus the injected carrier density. The carrier density N h = N e varies from 10 16 cm 3 to 10 19  cm 3 .

Fig. 10
Fig. 10

3D model for a linear OLWA. Notice that the lateral view is as in the 2D model.

Fig. 11
Fig. 11

Comparison of the far-field radiation patterns in the xy plane of 2D and 3D optical leaky wave antennas. In both cases a narrow radiating beam is obtained.

Equations (12)

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E ( x , y ) = n = E n ( y ) e i k x , n x ,      k x , n = k x , 0 + 2 n π / d ,
k x , 1 = β 1 + i α ,   with   β 1 = β 2 π / d ,
E ( x ) = E 1 e i β 1 x e α x        ( 0 < x < L ) ,
E F F ( ϕ ) = E 1 0 L e i β 1 x e α x e i k h cos ϕ x d x = E 1 e i ( k h cos ϕ β 1 i α ) L 1 i ( k h cos ϕ β 1 i α )
F ( ϕ ) = | E F F ( ϕ ) | | E 1 | = ( 1 + e 2 α L 2 e α L cos [ ( k h cos ϕ β 1 ) L ] ( k h cos ϕ β 1 ) 2 + α 2 ) 1 2 ,
Δ ϕ 3 dB 2 α k h
A F ( ϕ ) = n = 0 N 1 e i ( β 1 + i α ) n d e i ( k h cos ϕ ) n d = 1 e i ( k h cos ϕ β 1 i α ) N d 1 e i ( k h cos ϕ β 1 i α ) d ,
| A F ( ϕ ) | = ( 1 + e 2 α L 2 e α L cos [ ( k h cos ϕ β 1 ) L ] 1 + e 2 α d 2 e α d cos [ ( k h cos ϕ β 1 ) d ] ) 1 2 ,
D = 2 π | E max F F | 2 0 2 π | E F F ( ϕ ) | 2 d ϕ ,
Δ n S i ( N e , N h ) = ( 8.8 × 10 4 N e + 8.5 N h 0.8 ) × 10 18 ,
Δ α S i ( N e , N h ) = ( 8.5 N e + 6.0 N h ) × 10 16 ,
Q = F ( ϕ max , Δ n s i ) F ( ϕ max , 0 ) = α 1 e α L ( 1 + e 2 α L 2 e α L cos [ ( f f Δ n S i k 0 ) L ] ( f f Δ n S i k 0 ) 2 + α 2 ) 1 2 ,

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