Abstract

The mode expansion approach in vectorial form, using a complete set of guided modes of a circular step-index fiber (SIF), is developed and applied to analyze multimode interference in multimode fibers (MMFs) for the first time, to the best of our knowledge. The complete set of guided modes of an SIF is defined based on its modal properties, and a suitable modal orthogonality relation is identified to evaluate the coefficients in a mode expansion. An algorithm, adaptive to incident fields, is then developed to systematically and efficiently perform mode expansion in highly MMFs. The mode expansion approach is successfully applied to investigate the mode-selection properties of coreless fiber segments incorporated in multicore fiber lasers and the self-imaging in MMFs.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2007 (1)

2006 (4)

2005 (3)

2004 (2)

2003 (2)

A. Mehta, W. S. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

P.-L. Liu and S. De, "Fiber design--from optical mode to index profile," Opt. Eng. (Bellingham) 42, 981-984 (2003).
[CrossRef]

2000 (1)

1995 (1)

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

1994 (1)

A. Hardy and M. Ben-Artzi, "Expansion of an arbitrary field in terms of waveguide modes," IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

1992 (1)

1991 (1)

P. R. McIsaac, "Mode orthogonality in reciprocal and nonreciprocal waveguides," IEEE Trans. Microwave Theory Tech. 39, 1808-1816 (1991).
[CrossRef]

1975 (1)

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

1973 (1)

1961 (1)

1959 (1)

A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Electron Devices MTT-7, 441-446 (1959).

1958 (1)

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties of modes in passive and active uniform wave guides," J. Appl. Phys. 29, 794-799 (1958).
[CrossRef]

1957 (1)

L. R. Walker, "Orthogonality relation for gyrotropic waveguides," J. Appl. Phys. 28, 377 (1957).
[CrossRef]

Adler, R. B.

R. B. Adler, "Properties of guided waves on inhomogeneous cylindrical structures," Research Laboratory of Electronics, Massachusetts Institute of Technology, Tech. Rep. 102 (1949).

Alvarez-Chavez, J.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985).

Ben-Artzi, M.

A. Hardy and M. Ben-Artzi, "Expansion of an arbitrary field in terms of waveguide modes," IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

Bennett, C. R.

Bresler, A. D.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties of modes in passive and active uniform wave guides," J. Appl. Phys. 29, 794-799 (1958).
[CrossRef]

Brio, M.

K. M. Gundu, M. Brio, and J. V. Moloney, "A mixed high-order vector finite element method for waveguides: convergence and spurious mode studies," Int. J. Numer. Model. 18, 351-364 (2005).
[CrossRef]

Broeng, J.

Bryngdahl, O.

Burke, J. J.

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

Chen, S.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE, 1991).

De, S.

P.-L. Liu and S. De, "Fiber design--from optical mode to index profile," Opt. Eng. (Bellingham) 42, 981-984 (2003).
[CrossRef]

Farrell, G.

Q. Wang and G. Farrell, "Numerical investigation of multimode interference in a multimode fiber and its applications in optical sensing," Proc. SPIE 6189, 61891N (2006).
[CrossRef]

Q. Wang and G. Farrell, "All-fiber multimode-interference-based refractometer sensor: proposal and design," Opt. Lett. 31, 317-319 (2006).
[CrossRef] [PubMed]

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Fisher, D.

Glas, P.

Gu, X.

Gundu, K. M.

K. M. Gundu, M. Brio, and J. V. Moloney, "A mixed high-order vector finite element method for waveguides: convergence and spurious mode studies," Int. J. Numer. Model. 18, 351-364 (2005).
[CrossRef]

K. M. Gundu, College of Optical Sciences, the University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721, USA (personal communication, 2006).

Hadley, G. R.

Hardy, A.

A. Hardy and M. Ben-Artzi, "Expansion of an arbitrary field in terms of waveguide modes," IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

Johnson, E.

Johnson, E. G.

W. S. Mohammed, A. Mehta, and E. G. Johnson, "Wavelength tunable fiber lens based on multimode interference," J. Lightwave Technol. 22, 469-477 (2004).
[CrossRef]

A. Mehta, W. S. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

Joshi, G. H.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties of modes in passive and active uniform wave guides," J. Appl. Phys. 29, 794-799 (1958).
[CrossRef]

Kapany, N. S.

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

Leitner, M.

Li, H.

Li, L.

LiKamWa, P.

Liu, P.-L.

P.-L. Liu and S. De, "Fiber design--from optical mode to index profile," Opt. Eng. (Bellingham) 42, 981-984 (2003).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Mafi, A.

Marcuvitz, N.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties of modes in passive and active uniform wave guides," J. Appl. Phys. 29, 794-799 (1958).
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Martinez-Rios, A.

May-Arrioja, D.

McIsaac, P. R.

P. R. McIsaac, "Mode orthogonality in reciprocal and nonreciprocal waveguides," IEEE Trans. Microwave Theory Tech. 39, 1808-1816 (1991).
[CrossRef]

Mehta, A.

Michaille, L.

Mohammed, W. S.

Moloney, J. V.

Napartovich, A. P.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Petersson, A.

Peyghambarian, N.

Schülzgen, A.

Selvas, R.

Shepherd, T. J.

Simonsen, H. R.

Smith, P. W. E.

Snitzer, E.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Taylor, D. M.

Temyanko, V. L.

Torres-Gomez, I.

Ulrich, R.

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Villeneuve, A. T.

A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Electron Devices MTT-7, 441-446 (1959).

Vysotsky, D. D.

Walker, L. R.

L. R. Walker, "Orthogonality relation for gyrotropic waveguides," J. Appl. Phys. 28, 377 (1957).
[CrossRef]

Wang, Q.

Q. Wang and G. Farrell, "Numerical investigation of multimode interference in a multimode fiber and its applications in optical sensing," Proc. SPIE 6189, 61891N (2006).
[CrossRef]

Q. Wang and G. Farrell, "All-fiber multimode-interference-based refractometer sensor: proposal and design," Opt. Lett. 31, 317-319 (2006).
[CrossRef] [PubMed]

Wrage, M.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, 1997).

IEE Proc.: Optoelectron. (1)

A. Hardy and M. Ben-Artzi, "Expansion of an arbitrary field in terms of waveguide modes," IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Mehta, W. S. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, "Mode orthogonality in reciprocal and nonreciprocal waveguides," IEEE Trans. Microwave Theory Tech. 39, 1808-1816 (1991).
[CrossRef]

Int. J. Numer. Model. (1)

K. M. Gundu, M. Brio, and J. V. Moloney, "A mixed high-order vector finite element method for waveguides: convergence and spurious mode studies," Int. J. Numer. Model. 18, 351-364 (2005).
[CrossRef]

IRE Trans. Electron Devices (1)

A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Electron Devices MTT-7, 441-446 (1959).

J. Appl. Phys. (2)

L. R. Walker, "Orthogonality relation for gyrotropic waveguides," J. Appl. Phys. 28, 377 (1957).
[CrossRef]

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties of modes in passive and active uniform wave guides," J. Appl. Phys. 29, 794-799 (1958).
[CrossRef]

J. Lightwave Technol. (2)

W. S. Mohammed, A. Mehta, and E. G. Johnson, "Wavelength tunable fiber lens based on multimode interference," J. Lightwave Technol. 22, 469-477 (2004).
[CrossRef]

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

Opt. Eng. (Bellingham) (1)

P.-L. Liu and S. De, "Fiber design--from optical mode to index profile," Opt. Eng. (Bellingham) 42, 981-984 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Proc. SPIE (1)

Q. Wang and G. Farrell, "Numerical investigation of multimode interference in a multimode fiber and its applications in optical sensing," Proc. SPIE 6189, 61891N (2006).
[CrossRef]

Other (11)

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

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, 1997).

R. B. Adler, "Properties of guided waves on inhomogeneous cylindrical structures," Research Laboratory of Electronics, Massachusetts Institute of Technology, Tech. Rep. 102 (1949).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

R. E. Collin, Field Theory of Guided Waves (IEEE, 1991).

http://www.ece.byu.edu/photonics/connectors.parts/smf28.pdf.

K. M. Gundu, College of Optical Sciences, the University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721, USA (personal communication, 2006).

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

G. Arfken, Mathematical Methods for Physicists (Academic, 1985).

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Govternment Printing Office, 1972).

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Figures (10)

Fig. 1
Fig. 1

Two guided supermodes of the 19-core MCF: (a) HE 11 -like fundamental supermode and (b) TM 01 -like higher-order supermode.

Fig. 2
Fig. 2

Amplitude profiles of the field at distances of (a) 300, (b) 500, and (c) 1000 μ m along the coreless fiber, respectively. The fundamental supermode shown in Fig. 1a is the incident field and the FE-BPM is used to obtain the results.

Fig. 3
Fig. 3

Amplitude profiles of the field at distances of (a) 300, (b) 500, and (c) 1000 μ m along the coreless fiber, respectively. The fundamental supermode shown in Fig. 1a is the incident field and the mode expansion approach is employed here.

Fig. 4
Fig. 4

Effective amplitude reflection coefficient γ of (a) fundamental supermode in Fig. 1a or (b) higher-order supermode in Fig. 1b as a function of the propagation distance z along the OD 200 μ m coreless fiber or a bulk medium of the same refractive index as that of the coreless fiber, respectively.

Fig. 5
Fig. 5

Coefficients γ of all guided supermodes of the MCF laser at the propagation distances of 3320 and 6760 μ m along the coreless fiber segment.

Fig. 6
Fig. 6

(a) HE 11 mode of the SMF segment of the SMF-MMF-SMF structure and (b) the amplitude of the field at the self-imaging position along the MMF segment of the perfectly aligned SMF-MMF-SMF structure.

Fig. 7
Fig. 7

Coefficient γ of the field in the MMF segment of the SMF-MMF-SMF structure as a function of the propagation distance z. The solid curves are for the perfectly aligned structure, and the dash-dot curves are for the misaligned structure. The dashed curve corresponds to the coefficient γ of the field in a bulk medium of the same refractive index as that of the MMF core.

Fig. 8
Fig. 8

Coefficient γ at the self-imaging position as a function of the offset between the centers of the input SMF and MMF segments.

Fig. 9
Fig. 9

Graphical representation of Eqs. (2d, 2e).

Fig. 10
Fig. 10

Schematic of the cross section of a generalized SIF.

Tables (3)

Tables Icon

Table 1 Normalized Orthogonality Coefficients N m n Based on OR I

Tables Icon

Table 2 Normalized Orthogonality Coefficients N m n Based on OR II

Tables Icon

Table 3 Parameters for the SMF-MMF-SMF Structure at the Wavelength of 1.55 μ m

Equations (87)

Equations on this page are rendered with MathJax. Learn more.

E ( r , ϕ , z , t ) = E ( r ) exp ( i l ϕ ) exp [ i ( ω t β z ) ] ,
H ( r , ϕ , z , t ) = H ( r ) exp ( i l ϕ ) exp [ i ( ω t β z ) ] ,
J l + 1 ( h a ) h a J l ( h a ) = n 1 2 + n 2 2 2 n 1 2 K l ( q a ) q a K l ( q a ) + ( l ( h a ) 2 R ) ,
J l 1 ( h a ) h a J l ( h a ) = n 1 2 + n 2 2 2 n 1 2 K l ( q a ) q a K l ( q a ) + ( l ( h a ) 2 R ) ,
R = [ ( n 1 2 n 2 2 2 n 1 2 ) 2 ( K l ( q a ) q a K l ( q a ) ) 2 + ( l β n 1 k 0 ) 2 ( 1 ( q a ) 2 + 1 ( h a ) 2 ) ] 1 2 ,
h 2 = n 1 2 k 0 2 β 2 , q 2 = β 2 n 2 2 k 0 2 ,
V 2 = ( h a ) 2 + ( q a ) 2 = ( n 1 2 n 2 2 ) k 0 2 a 2 ,
lim h a V RHS HE ( h a ) = + for l = 1 ,
lim h a V RHS HE ( h a ) = 1 l 1 n 2 2 n 1 2 + n 2 2 for l > 1 .
J 1 ( V m ) = 0 , V 1 = 0 ,
J l 1 ( V m ) V m J l ( V m ) = 1 l 1 n 2 2 n 1 2 + n 2 2 , V m 0 ,
P = l ( 1 q 2 a 2 + 1 h 2 a 2 ) ( J l ( h a ) h a J l ( h a ) + K l ( q a ) q a K l ( q a ) ) 1
E r ( r , ϕ ) = i β 2 h A [ ( 1 P ) J l 1 ( h r ) ( 1 + P ) J l + 1 ( h r ) ] exp ( i l ϕ ) ,
E ϕ ( r , ϕ ) = β 2 h A [ ( 1 P ) J l 1 ( h r ) + ( 1 + P ) J l + 1 ( h r ) ] exp ( i l ϕ ) ,
E z ( r , ϕ ) = A J l ( h r ) exp ( i l ϕ ) ,
H r ( r , ϕ ) = ω ε 1 2 h A [ ( f 1 P 1 ) J l 1 ( h r ) ( f 1 P + 1 ) J l + 1 ( h r ) ] exp ( i l ϕ ) ,
H ϕ ( r , ϕ ) = i ω ε 1 2 h A [ ( f 1 P 1 ) J l 1 ( h r ) + ( f 1 P + 1 ) J l + 1 ( h r ) ] exp ( i l ϕ ) ,
H z ( r , ϕ ) = i ω ε 1 β A f 1 P J l ( h r ) exp ( i l ϕ ) ,
f 1 = β 2 ( n 1 2 k 0 2 ) ;
E r ( r , ϕ ) = i β 2 q C [ ( 1 P ) K l 1 ( q r ) + ( 1 + P ) K l + 1 ( q r ) ] exp ( i l ϕ ) ,
E ϕ ( r , ϕ ) = β 2 q C [ ( 1 P ) K l 1 ( q r ) ( 1 + P ) K l + 1 ( q r ) ] exp ( i l ϕ ) ,
E z ( r , ϕ ) = C K l ( q r ) exp ( i l ϕ ) ,
H r ( r , ϕ ) = ω ε 2 2 q C [ ( 1 f 2 P ) K l 1 ( q r ) ( 1 + f 2 P ) K l + 1 ( q r ) ] exp ( i l ϕ ) ,
H ϕ ( r , ϕ ) = i ω ε 2 2 q C [ ( 1 f 2 P ) K l 1 ( q r ) + ( 1 + f 2 P ) K l + 1 ( q r ) ] exp ( i l ϕ ) ,
H z ( r , ϕ ) = i ω ε 2 β C f 2 P K l ( q r ) exp ( i l ϕ ) ,
C = A J l ( h a ) K l ( q a ) ,
f 2 = β 2 ( n 2 2 k 0 2 ) .
E ϕ ( r , ϕ ) = i β h A J 1 ( h r ) ,
H r ( r , ϕ ) = i ω ε 0 β 2 h k 0 2 A J 1 ( h r ) ,
H z ( r , ϕ ) = ω ε 0 β k 0 2 A J 0 ( h r ) ,
E ϕ ( r , ϕ ) = i β q A J 0 ( h a ) K 0 ( q a ) K 1 ( q r ) ,
H r ( r , ϕ ) = i ω ε 0 β 2 q k 0 2 A J 0 ( h a ) K 0 ( q a ) K 1 ( q r ) ,
H z ( r , ϕ ) = ω ε 0 β k 0 2 A J 0 ( h a ) K 0 ( q a ) K 0 ( q r ) ,
E ( + l ) ( r , ϕ ) = E ( r , ϕ ; l ) = E ( l ) ( r ) exp ( i l ϕ ) ,
E ( l ) ( r , ϕ ) = E ( r , ϕ ; l ) = E ( l ) ( r ) exp ( i l ϕ ) .
E r ( l ) ( r ) = [ E r ( l ) ( r ) ] * , E ϕ ( l ) ( r ) = [ E ϕ ( l ) ( r ) ] * , E z ( l ) ( r ) = [ E z ( l ) ( r ) ] * ,
E ¯ ( n ) ( x , y , z , t ) = ( e T ( n ) ( x , y ) + z ̂ e z ( n ) ( x , y ) ) exp ( γ n z ) exp ( i ω t ) ,
S d s z ̂ ( e T ( n ) × h T ( m ) ) = N n δ m n ,
S d s z ̂ ( e T ( n ) × h T ( m ) * ) = 2 P n δ m n
S d s e T ( n ) e T ( m ) = M n δ m n ,
S d s e T ( n ) e T ( m ) * = Q n δ m n ,
N m n = P m n [ P m m P n n ] 1 2 ,
P m n ( I ) = S d s z ̂ ( e T ( m ) × h T ( n ) * ) ,
P m n ( I I ) = S d s e T ( m ) e T ( n ) *
E ( r , ϕ , z ) = n = 1 N C n E ( n ) ( r , ϕ , z ) = n = 1 N C n e ( n ) ( r , ϕ ) exp ( i β n z ) ,
C n = S d s z ̂ [ E T i n ( r , ϕ , z = 0 ) × h T ( n ) * ( r , ϕ ) ] S d s z ̂ ( e T ( n ) ( r , ϕ ) × h T ( n ) * ( r , ϕ ) ) ,
E T i n ( r , ϕ , z = 0 ) = l = L L e T i n ( r ; l ) exp ( i l ϕ ) ,
H T i n ( r , ϕ , z = 0 ) = l = L L h T i n ( r ; l ) exp ( i l ϕ ) ,
S d s z ̂ [ E T i n × H T i n * ] = l = L L S d s z ̂ [ e T i n ( r ; l ) × h T i n * ( r ; l ) ] ,
R z = z ̂ ( E T i n × H T i n * ) 2 ,
E T i n ( r , ϕ , z = 0 ) E T ( s ) ( r , ϕ , z = 0 ) = l S L e T i n ( r ; l ) exp ( i l ϕ ) ,
E T ( r , ϕ , z ) = l = L L n l = 1 N l C l , n l e T ( l , n l ) ( r ) exp [ i ( l ϕ β l , n l z ) ] ,
C l , n l = 0 r d r z ̂ [ e T i n ( r ; l ) × h T ( l , n l ) * ( r ) ] 0 r d r z ̂ [ e T ( l , n l ) ( r ) × h T ( l , n l ) * ( r ) ] .
E T ( r , ϕ , z ) l S L n l S N l C l , n l e T ( l , n l ) ( r ) exp [ i ( l ϕ β l , n l z ) ] .
γ ( z ) = S d s z ̂ [ E T ( r , ϕ , z ) × H T * ( r , ϕ , z = 0 ) ] S d s z ̂ [ E T ( r , ϕ , z = 0 ) × H T * ( r , ϕ , z = 0 ) ] .
γ ( z ) = l S L n l S N l C l , n l 2 P ( l , n l ) exp ( i β l , n l z ) T P ,
T P = l S L n l S N l C l , n l 2 P ( l , n l ) ,
P ( l , n l ) = 0 r d r z ̂ [ e T ( l , n l ) ( r ) × h T ( l , n l ) * ( r ) ] .
RHS EH ( h a ) = R A + R B R ,
R A = n 1 2 + n 2 2 2 n 1 2 K l ( q a ) q a K l ( q a ) = n 1 2 + n 2 2 2 n 1 2 ( l ( q a ) 2 + K l 1 ( q a ) q a K l ( q a ) ) > 0 ,
R B = l ( h a ) 2 .
RHS EH ( h a ) = R A + R B R < R B R < l ( h a ) 2 l β n 1 k 0 ( 1 ( q a ) 2 + 1 ( h a ) 2 ) = ( 1 sin 2 ϕ 1 + cos θ ) l q 2 a 2 < 0 .
lim h a 0 RHS EH ( h a ) = lim h a 0 ( R A + R B R ) ( R A + R B + R ) ( R A + R B + R ) = n 1 2 + n 2 2 2 n 1 2 ( 2 l V 2 + K l 1 ( V ) V K l ( V ) ) ,
lim h a V RHS EH ( h a ) < lim h a V ( R A ) = lim q a 0 ( R A ) .
RHS HE ( h a ) = R A + R B R .
RHS HE ( h a ) = R A + R B R = I ( R A + R B + R ) ,
I = ( R A + R B R ) ( R A + R B + R ) = I 1 + I 2 ,
I 1 = l 2 ( h a ) 4 ( l β n 1 k 0 ) 2 ( 1 q 2 a 2 + 1 h 2 a 2 ) + ( 1 + n 2 2 n 1 2 ) l 2 ( h a ) 2 ( q a ) 2 + n 2 2 n 1 2 l 2 ( q a ) 4
= l 2 ( q a ) 4 [ ( 2 sin 2 ϕ sin 2 θ 1 sin 4 ϕ sin 2 θ ) + ( sin 2 ϕ sin 2 θ 1 ) ( 2 sin 2 ϕ ) + ( 1 sin 2 ϕ ) ] = 0 ,
I 2 = K l 1 ( q a ) q a K l ( q a ) [ l ( h a ) 2 + n 2 2 n 1 2 ( l ( h a ) 2 + 2 l ( q a ) 2 + K l 1 ( q a ) q a K l ( q a ) ) ] > 0 .
lim h a 0 RHS HE ( h a ) = lim h a 0 I ( R A + R B + R ) = n 1 2 + n 2 2 2 n 1 2 K l 1 ( V ) V K l ( V ) ,
lim h a V RHS HE ( h a ) = lim q a 0 I ( R A + R B + R )
= lim q a 0 2 n 2 2 n 1 2 + n 2 2 ln ( q a ) + for l = 1 ,
lim h a V RHS HE ( h a ) = 1 l 1 n 2 2 n 1 2 + n 2 2 for l > 1 .
K 0 ( x ) ( x K 1 ( x ) ) ln x ,
K l 1 ( x ) ( x K l ( x ) ) 1 ( 2 ( l 1 ) )
( E ( n ) × H ( m ) E ( m ) × H ( n ) ) = 0 ,
E ( n ) = E T ( n ) + E z ( n ) z ̂ = ( e T ( n ) ( x , y ) + e z ( n ) ( x , y ) z ̂ ) exp ( Γ n z ) ,
H ( n ) = H T ( n ) + H z ( n ) z ̂ = ( h T ( n ) ( x , y ) + h z ( n ) ( x , y ) ) exp ( Γ n z ) ,
T ( E ( n ) × H ( m ) E ( m ) × H ( n ) ) = ( Γ n + Γ m ) z ̂ ( E ( n ) × H ( m ) E ( m ) × H ( n ) ) .
( Γ n + Γ m ) s 1 + s 2 d S ( E ( n ) × H ( m ) E ( m ) × H ( n ) ) z ̂ = C d l ( E ( n ) H ( m ) E ( m ) × H ( n ) ) ( 1 ) ( E ( n ) × H ( m ) E ( m ) × H ( n ) ) ( 2 ) n ̂ ,
( E ( n ) × H ( m ) ) n ̂ = ( E T ( n ) × H z ( m ) z ̂ + E z ( n ) z ̂ × H T ( m ) ) n ̂ ,
R S = C d l [ ( E T ( n ) ( 1 ) E T ( n ) ( 2 ) ) H z ( m ) ( E T ( m ) ( 1 ) E T ( m ) ( 2 ) ) H z ( n ) ] × z ̂ n ̂ .
E T ( n ) ( 1 ) E T ( n ) ( 2 ) = ( E T N ( n ) ( 1 ) E T N ( n ) ( 2 ) ) n ̂ ,
R S = C d l [ ( E T N ( n ) ( 1 ) E T N ( n ) ( 2 ) ) n ̂ H z ( m ) ( E T N ( m ) ( 1 ) E T N ( m ) ( 2 ) ) n ̂ H z ( n ) ] × z ̂ n ̂ = 0 .
( Γ n + Γ m ) s 1 + s 2 d S ( e T ( n ) × h T ( m ) e T ( m ) × h T ( n ) ) z ̂ = 0 .
( E ( n ) × H ( m ) * + E ( m ) * × H ( n ) ) = 0 ,

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