Abstract

Light transfer between a curved single-mode fiber and a two-mode planar waveguide is investigated experimentally with respect to its dependence on the refractive index and the thickness of the planar waveguide and for different polishing depths. The behavior of this distributed coupler is treated theoretically with a coupled-mode model, which takes into account the two dimensions of the waveguide configuration, as well as the even or the odd symmetry of the planar waveguide mode field. Coupling of a fiber to a multimode planar waveguide is also considered.

© 1994 Optical Society of America

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References

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  1. C. Millar, M. Brierley, and S. Mallinson, “Exposed-core single-mode-fiber channel-dropping filter using a high-index overlay waveguide,” Opt. Lett. 12, 284–286 (1987).
    [CrossRef] [PubMed]
  2. M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
    [CrossRef]
  3. W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
    [CrossRef]
  4. W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
    [CrossRef]
  5. A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
    [CrossRef]
  6. K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
    [CrossRef]
  7. A. Andreev and K. Panajotov, “Distributed single-mode fiber-to-single-mode planar waveguide coupler,” IEEE J. Lightwave Technol. (to be published).
  8. A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.
  9. D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” IEEE J. Lightwave Technol. 7, 122–130 (1989).
    [CrossRef]
  10. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  11. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  12. O. Leminger and R. Zengerle, “Determination of single-mode fiber coupler design parameters from loss measurements,” J. Lightwave Technol. 3, 864–867 (1985).
    [CrossRef]
  13. Cargille refractive-index liquids, Cargille, Inc., Cedar Grove, N.J. 07009–1289.
  14. M. Fox, “Calculation of equivalent step-index parameters for single-mode fiber,” Opt. Quantum Electron. 15, 451–455 (1983).
    [CrossRef]

1992 (3)

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
[CrossRef]

K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
[CrossRef]

1991 (1)

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

1990 (1)

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

1989 (1)

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” IEEE J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

1987 (1)

1985 (1)

O. Leminger and R. Zengerle, “Determination of single-mode fiber coupler design parameters from loss measurements,” J. Lightwave Technol. 3, 864–867 (1985).
[CrossRef]

1983 (1)

M. Fox, “Calculation of equivalent step-index parameters for single-mode fiber,” Opt. Quantum Electron. 15, 451–455 (1983).
[CrossRef]

1971 (1)

Andreev, A.

A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
[CrossRef]

K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
[CrossRef]

A. Andreev and K. Panajotov, “Distributed single-mode fiber-to-single-mode planar waveguide coupler,” IEEE J. Lightwave Technol. (to be published).

A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.

Brierley, M.

Croituro, N.

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

Culshaw, B.

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Fox, M.

M. Fox, “Calculation of equivalent step-index parameters for single-mode fiber,” Opt. Quantum Electron. 15, 451–455 (1983).
[CrossRef]

Gill, M.

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Gloge, D.

Hardy, A.

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

Iztkovich, M.

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

Johnstone, W.

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Koprinarova, J.

A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.

Leminger, O.

O. Leminger and R. Zengerle, “Determination of single-mode fiber coupler design parameters from loss measurements,” J. Lightwave Technol. 3, 864–867 (1985).
[CrossRef]

Mallinson, S.

Marcuse, D.

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” IEEE J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

McDonach, A.

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Millar, C.

Moodie, D.

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Murray, S.

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Panajotov, K.

A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
[CrossRef]

K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
[CrossRef]

A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.

A. Andreev and K. Panajotov, “Distributed single-mode fiber-to-single-mode planar waveguide coupler,” IEEE J. Lightwave Technol. (to be published).

Thursby, G.

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

Tur, M.

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

Varshey, R.

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

Zafirova, B.

A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
[CrossRef]

K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
[CrossRef]

A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.

Zengerle, R.

O. Leminger and R. Zengerle, “Determination of single-mode fiber coupler design parameters from loss measurements,” J. Lightwave Technol. 3, 864–867 (1985).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (3)

M. Iztkovich, M. Tur, A. Hardy, and N. Croituro, “In-situ investigation of coupling between a fiber and a slab waveguide,” Electron. Lett. 26, 1104–1105 (1990).
[CrossRef]

W. Johnstone, G. Thursby, D. Moodie, R. Varshey, and B. Culshaw, “Fibre optic wavelength channel selector with high resolution,” Electron. Lett. 28, 1364–1365 (1992).
[CrossRef]

W. Johnstone, S. Murray, G. Thursby, M. Gill, A. McDonach, D. Moodie, and B. Culshaw, “Fibre optic modulators using active multimode waveguide overlays,” Electron. Lett. 27, 894–896 (1991).
[CrossRef]

IEEE J. Lightwave Technol. (3)

A. Andreev, K. Panajotov, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part I,” IEEE J. Lightwave Technol. 10, 882–887 (1992).
[CrossRef]

K. Panajotov, A. Andreev, and B. Zafirova, “Polished fiber-optic coupler with intermediate planar waveguiding layer—part II,” IEEE J. Lightwave Technol. 10, 1363–1366 (1992).
[CrossRef]

D. Marcuse, “Investigation of coupling between a fiber and an infinite slab,” IEEE J. Lightwave Technol. 7, 122–130 (1989).
[CrossRef]

J. Lightwave Technol. (1)

O. Leminger and R. Zengerle, “Determination of single-mode fiber coupler design parameters from loss measurements,” J. Lightwave Technol. 3, 864–867 (1985).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Fox, “Calculation of equivalent step-index parameters for single-mode fiber,” Opt. Quantum Electron. 15, 451–455 (1983).
[CrossRef]

Other (4)

Cargille refractive-index liquids, Cargille, Inc., Cedar Grove, N.J. 07009–1289.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

A. Andreev and K. Panajotov, “Distributed single-mode fiber-to-single-mode planar waveguide coupler,” IEEE J. Lightwave Technol. (to be published).

A. Andreev, K. Panajotov, B. Zafirova, and J. Koprinarova, “Single-mode fiber to planar waveguide coupling,” presented at the Fiber Optics Networks and Video Communications Symposium, Berlin, 1993.

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Figures (8)

Fig. 1
Fig. 1

Longitudinal and cross-sectional views of the distributed fiber-to-planar-waveguide coupler.

Fig. 2
Fig. 2

Propagation constants of the even mode (solid curve) and the odd mode (dashed curve) of the planar waveguide, calculated from Eqs. (8) and (12) as a function of the planar-waveguide refractive index. The fiber propagation constant, calculated from Eq. (4), is presented as the horizontal line.

Fig. 3
Fig. 3

Dependence of the fiber output power Pf, taken as |a|2, on the planar-waveguide refractive index ns for different planar-waveguide thicknesses and polishing depths. d = 2.2 μm: (a) s0 = 3.706 μm, (b) s0 = 1.946 μm, (c) s0 = 0.466 μm. d = 1.7 μm: (d) s0 = 3.706 μm, (e) s0 = 1.946 μm. The experimental results are denoted by dots; the theoretical ones, calculated from Eqs. (25) and (26), with solid lines.

Fig. 4
Fig. 4

Even-mode coupling coefficients (a) Kfeν and (b) K as a function of the transverse order ν and the planar-waveguide refractive index ns at z = 0 for s(z) = s0 = 0.466 μm and d = 2.2 μm.

Fig. 5
Fig. 5

Odd-mode coupling coefficients (a) Kfsν and (b) K as a function of the transverse order ν and the planar-waveguide refractive index ns at z = 0 for s(z) = s0 = 0.466 μm and d = 2.2 μm.

Fig. 6
Fig. 6

(a) Even-mode (solid curves) and odd-mode (dashed curves) propagation constants of the planar waveguide, calculated from Eqs. (8) and (12). (b) Dependence of the fiber output power Pf = |a|2, calculated from Eqs. (25) and (26), on the planar-waveguide refractive index ns for a multimode planar waveguide with thickness d = 10 μm and polishing depth s0 = 1.946 μm.

Fig. 7
Fig. 7

Even-mode planar-waveguide optical power distribution P(x, 0, z) = |Ψ(x, 0, z)|2 for d = 2.2 μm, s0 = 1.946 μm; (a) ns = 1.4578, (b) ns = 1.4586.

Fig. 8
Fig. 8

Odd-mode planar-waveguide optical power distribution P(x, 0, z) = |Ψ(x, 0, z)|2 for d = 2.2 μm, s0 = 1.946 μm; (a) ns = 1.4733, (b) ns = 1.4755.

Equations (29)

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t 2 F + ( k 2 n f 2 - β f 2 ) F = 0 , t 2 E ν + ( k 2 n s 2 - β e ν 2 ) E ν = 0 , ν = 1 , 2 , , N e , t 2 S ν + ( k 2 n s 2 - β s ν 2 ) S ν = 0 , ν = 1 , 2 , , N s ,
F = γ f π V f 2 J 1 ( k f a ) { J 0 ( k f r ) for r a J 0 ( k f a ) K 0 ( γ f a ) K 0 ( γ f r ) for r > a ,
k f 2 = k 2 n f 2 - β f 2 = V f 2 / a 2 - γ f 2 .
k f J 1 ( k f a ) J 0 ( k f a ) = γ f K 1 ( γ f a ) K 0 ( γ f a ) .
E ν = [ γ e ( 1 + γ e d / 2 ) D ] 1 / 2 × { cos ( k e y ) cos ( σ e ν x ) for y d / 2 cos ( k e d / 2 ) exp [ - γ e ( y - d / 2 ) ] cos ( σ e ν x ) , for y > d / 2 ,
k e 2 = k 2 n s 2 - β e 2 = 4 V s 2 - γ e 2 ,
β e ν 2 = β e 2 - σ e ν 2 = β e 2 - ( ν + 1 2 ) 2 π 2 D 2             for ν = 0 , 1 , 2 , , N e .
tan ( k e d 2 ) = γ e k e .
S ν = [ γ s ( 1 + γ s d / 2 ) D ] 1 / 2 × { sin ( k s y ) cos ( σ s ν x ) for y d / 2 sin ( k s d / 2 ) exp [ - γ s ( y - d / 2 ) ] cos ( σ s ν x ) for y > d / 2 ,
k s 2 = k 2 n s 2 - β s 2 = 4 V s 2 - γ s 2 ,
β s ν 2 = β s 2 - σ s ν 2 = β s 2 - ( ν + 1 2 ) 2 π 2 D 2             for ν = 0 , 1 , 2 , , N s .
cot ( k s d 2 ) = - γ s k s .
Ψ = a ( z ) F exp ( - j β f z ) + ν = 0 N e b ν ( z ) E ν exp ( - j β e ν z ) + ν = 0 N s c ν ( z ) S ν exp ( - j β s ν z ) ,
t 2 Ψ + 2 Ψ z 2 + k 2 [ ( n f 2 - n 0 2 ) + ( n s 2 - n 0 2 ) - n 0 2 ] = 0.
d a d z = - j Q f a - j ν = 0 N e K f e ν b ν exp [ j ( β f - β e ν ) z ] - j μ = 0 N s K f s μ c μ exp [ j ( β f - β s μ ) z ] , d b ν d z = - J K e ν a exp [ j ( β e ν - β e μ ) z ] - j μ = 0 N e Q e ν μ b μ exp [ j ( β e ν - β e μ ) z ] - j μ = 0 N s R e ν μ c μ exp [ j ( β e ν - β s μ ) z ] , d c ν d z = - j K s ν a exp [ j ( β s ν - β f ) z ] - j μ = 0 N e R s ν μ b μ exp [ j ( β s ν - β e μ ) z ] - j μ = 0 N s Q s ν μ c μ exp [ j ( β s ν - β s μ ) z ] .
Q f = k 2 2 β f ( n s 2 - n 0 2 ) F 2 d x d y ,
Q e ν μ = k 2 2 β e ν ( n f 2 - n 0 2 ) E ν E μ d x d y , Q s ν μ = k 2 2 β s ν ( n f 2 - n 0 2 ) S ν S μ d x d y .
K f e ν = k 2 2 β f ( n f 2 - n 0 2 ) F E ν d x d y , K f s ν = k 2 2 β f ( n f 2 - n 0 2 ) F S ν d x d y ,
K e ν = k 2 2 β e ν ( n s 2 - n 0 2 ) F E ν d x d y , K s ν = k 2 2 β s ν ( n s 2 - n 0 2 ) F S ν d x d y ,
R e μ ν = k 2 2 β e μ ( n f 2 - n 0 2 ) E μ S ν d x d y , R s μ ν = k 2 2 β s μ ( n f 2 - n 0 2 ) S μ E ν d x d y .
s ( z ) s 0 + z 2 2 R .
2 a = 1.8198 w 0 c , Δ = Q / ( 1 + 2 Q ) , Q = 0.29297 ( λ c 2 a n 0 ) 2 .
d a d z = - j Q f a - j ν = 0 N e K f e ν exp [ j ( β f - β e ν ) z ] b ν , d b ν d z = - j μ = 0 N e Q e ν μ exp [ j ( β e ν - β e μ ) z ] b μ - j K s ν a exp [ j ( β e ν - β f ) z ] ,
d c ν d z = - j μ = 0 N s Q s ν μ exp [ j ( β s ν - β s μ ) z ] c μ , d c ν d z = - j μ = 0 N s Q s ν μ exp [ j ( β s ν - β s μ ) z ] c μ - j K s ν a exp [ j ( β s ν - β f ) z ] ,
d a d z = - j Q f a - j ν = 0 N e K f e ν exp [ j ( β f - β e ν ) z ] b ν , d b ν d z = - j K e ν a exp [ j ( β e ν - β f ) z ] ,
d a d z = - j Q f a - j μ = 0 N s K f s μ exp [ j ( β f - β s μ ) z ] c μ , d c ν d z = - j K s ν a exp [ j ( β s ν - β f ) z ] ,
K f e ν = γ f V f k e d 2 β f a V s [ π γ e ( 1 + γ e d / 2 ) D ] 1 / 2 exp [ - γ e ( a + s ) ] × ( γ e 2 - σ e ν 2 ) 1 / 2 J 0 ( k f a ) I 1 [ ( γ e 2 - σ e ν 2 ) 1 / 2 a ] + k f J 1 ( k f a ) I 0 [ ( γ e 2 - σ e ν 2 ) 1 / 2 a ] ( k f 2 + γ e 2 - σ e ν 2 ) J 1 ( k f a ) K e ν = k e γ f V s β e ν d V f [ π γ e ( 1 + γ e d / 2 ) D ] 1 / 2 J 0 ( k f a ) J 1 ( k f a ) K 0 ( γ f a ) exp [ - ( γ f 2 + σ e ν 2 ) 1 / 2 ( a + s ) ] × ( γ f 2 - σ e ν 2 ) 1 / 2 { 1 - exp [ - ( γ f 2 + σ e ν 2 ) 1 / 2 d ] } + γ s { 1 + exp [ - ( γ f 2 + σ e ν 2 ) 1 / 2 d ] } ( k e 2 + γ f 2 + σ e ν 2 ) ( γ f 2 + σ e ν 2 ) 1 / 2 , Q f = π V s 2 J 0 ( k f a ) { erf [ 2 γ f ( a + s + d ) ] 1 / 2 - erf [ 2 γ f ( a + s ) ] 1 / 2 } 2 β f d 2 V f 2 K 0 2 ( γ f a ) J 1 2 ( k f a ) , Q e μ ν = π k e 2 γ e V f 2 d 2 exp [ - 2 γ e ( a + s ) ] 8 β e μ D ( 1 + γ e d / 2 ) a V s 2 ( I 1 { [ 4 γ e 2 - ( σ e μ - σ e ν ) 2 ] 1 / 2 a } [ 4 γ e 2 - ( σ e μ - σ e ν ) 2 ] 1 / 2 + I 1 { [ 4 γ e 2 - ( σ e μ + σ e ν ) 2 ] 1 / 2 a } [ 4 γ e 2 - ( σ e μ + σ e ν ) 2 ] 1 / 2 ) .
K s ν = k s γ f V s β s ν d V f [ π γ s ( 1 + γ s d / 2 ) D ] 1 / 2 J 0 ( k f a ) J 1 ( k f a ) K 0 ( γ f a ) exp [ - ( γ f 2 + σ s ν 2 ) 1 / 2 ( a + s ) ] × ( γ f 2 - σ s ν 2 ) 1 / 2 { 1 + exp [ - ( γ f 2 + σ s ν 2 ) 1 / 2 d ] } + γ s { - 1 + exp [ - ( γ f 2 + σ s ν 2 ) 1 / 2 d ] } ( k s 2 + γ f 2 + σ s ν 2 ) ( γ f 2 + σ s ν 2 ) 1 / 2 .
R e μ ν = β e μ β s ν R s ν μ = π k e k s V f 2 d 2 exp [ - ( γ e + γ s ) ( a + s ) ] 8 β e μ a V s 2 D × [ γ e γ s ( 1 + γ e d / 2 ) ( 1 + γ s d / 2 ) ] 1 / 2 × ( I 1 { [ ( γ e + γ s ) 2 - ( σ s ν - σ e μ ) 2 ] 1 / 2 a } [ ( γ e + γ s ) 2 - ( σ s ν - σ e μ ) 2 ] 1 / 2 + I 1 { [ ( γ e + γ s ) 2 - ( σ s ν + σ e μ ) 2 ] 1 / 2 a } [ ( γ e + γ s ) 2 - ( σ s ν + σ e μ ) 2 ] 1 / 2 ) .

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