Abstract

A fully concatenated thermo-optical model is presented to predict the thermo-optical behavior of an intrinsically heated polymer fiber Bragg grating (PFBG). Coupled-mode theory and heat-conduction theory are first used to determine the axial heat generation and temperature distribution of a PFBG and the transfer matrix method (TMM) is subsequently employed to predict its thermo-optical behavior. The validity of the TMM is corroborated experimentally using an externally heated glass fiber Bragg grating (FBG) with an axially decaying temperature field. The verified model is utilized to investigate the thermo-optical behavior of a poly(methyl methacrylate) (PMMA) FBG. The counteracting thermally driven changes in the refractive index and the grating pitch, respectively, are found to be of comparable magnitude and to result in very modest net shifts in the Bragg wavelengths despite the considerable temperature changes induced by the absorption of the incident light.

© 2007 Optical Society of America

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References

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    [CrossRef]
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2005 (1)

2004 (1)

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

2003 (3)

2002 (2)

H. Y. Liu, G. D. Peng, and P. L. Chu, 'Thermal stability of gratings in PMMA and CYTOP polymer fibers,' Opt. Commun. 24, 151-156 (2002).

M. Zhou, 'Low-loss polymeric materials for passive waveguide components in fiber optical communication,' Opt. Eng. 41, 1631-1643 (2002).
[CrossRef]

2000 (2)

G. D. Peng and P. L. Chu, 'Polymer optical fiber photosensitivities and highly tunable fiber gratings,' Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

L. Eldada and L. W. Shacklette, 'Advances in polymer integrated optics,' IEEE J. Sel. Top. Quantum Electron. 6, 54-68 (2000).
[CrossRef]

1999 (1)

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

1997 (1)

T. Erdogan, 'Fiber grating spectra,' J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1994 (1)

1987 (2)

T. Kaino, 'Preparation of plastic optical fibers for near-IR region transmission,' J. Polym. Sci. Part A: Polym. Chem. 25, 37-46 (1987).
[CrossRef]

M. Yamada and K. Sakuda, 'Analysis of almost-periodic distributed slab waveguides via a fundamental matrix approach,' Appl. Opt. 26, 3474-3478 (1987).
[CrossRef] [PubMed]

1985 (1)

1973 (1)

A. Yariv, 'Coupled-mode theory for guided-wave optics,' IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, 'Coupled-wave theory of distributed feedback lasers,' J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, 'Coupled-wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

Bar-Cohen, A.

A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

Barton, J. S.

Beeson, K. W.

Bennion, I.

Chen, C.-L.

C.-L. Chen, Elements of Optoelectronics and Fiber Optics (Irwin, 1996).

Chu, P. L.

H. Y. Liu, G. D. Peng, and P. L. Chu, 'Thermal stability of gratings in PMMA and CYTOP polymer fibers,' Opt. Commun. 24, 151-156 (2002).

G. D. Peng and P. L. Chu, 'Polymer optical fiber photosensitivities and highly tunable fiber gratings,' Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

Colegrove, J.

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

Craig, D. M.

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

Cullen, M. R.

D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics (PWS-KENT, 1992).

Dobb, H.

Eldada, L.

L. Eldada and L. W. Shacklette, 'Advances in polymer integrated optics,' IEEE J. Sel. Top. Quantum Electron. 6, 54-68 (2000).
[CrossRef]

Erdogan, T.

T. Erdogan, 'Fiber grating spectra,' J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Fender, A.

Hall, D. G.

Jones, J. D. C.

Kaino, T.

T. Kaino, 'Preparation of plastic optical fibers for near-IR region transmission,' J. Polym. Sci. Part A: Polym. Chem. 25, 37-46 (1987).
[CrossRef]

Kalli, K.

A. Othonos and K. Kalli, Fiber Bragg Gratings--Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Kang, J.-W.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Kim, D. Y.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Kim, J.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Kim, J.-P.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Kim, K. J.

K. J. Kim, 'Thermo-structural influences on optical characteristics of polymer Bragg gratings,' Ph.D. dissertation (University of Maryland, 2006).

Kim, M.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, 'Coupled-wave theory of distributed feedback lasers,' J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

H. Kogelnik, 'Coupled-wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

Kostuk, R. K.

Kraus, A. D.

A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

Lee, J.-S.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Liu, H. Y.

H. Y. Liu, G. D. Peng, and P. L. Chu, 'Thermal stability of gratings in PMMA and CYTOP polymer fibers,' Opt. Commun. 24, 151-156 (2002).

MacPherson, W. N.

Moaveni, S.

S. Moaveni, Finite Element Analysis--Theory and Application with ANSYS (Prentice-Hall, 2002).

Othonos, A.

A. Othonos and K. Kalli, Fiber Bragg Gratings--Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Ozisik, M. N.

M. N. Ozisik, Heat Conduction (Wiley, 1980).

Peng, G. D.

H. Y. Liu, G. D. Peng, and P. L. Chu, 'Thermal stability of gratings in PMMA and CYTOP polymer fibers,' Opt. Commun. 24, 151-156 (2002).

G. D. Peng and P. L. Chu, 'Polymer optical fiber photosensitivities and highly tunable fiber gratings,' Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

Poladian, L.

Sakuda, K.

Sato, A.

Scepanovic, M.

Shacklette, L. W.

Shank, C. V.

H. Kogelnik and C. V. Shank, 'Coupled-wave theory of distributed feedback lasers,' J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

Silva-López, M.

Sipe, J. E.

Sterke, C. M. D.

Tang, S.

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

Tang, Y.

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

Webb, D. J.

Weber, M. J.

M. J. Weber, Handbook of Laser Science and Technology Supplement 2: Optical Materials (CRC, 1995).

Weller-Brophy, L. A.

Wu, B.

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

Xiong, Z.

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

Yamada, M.

Yariv, A.

A. Yariv, 'Coupled-mode theory for guided-wave optics,' IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Yoo, S.-J.

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Zhang, L.

Zhao, D.

Zhou, M.

M. Zhou, 'Low-loss polymeric materials for passive waveguide components in fiber optical communication,' Opt. Eng. 41, 1631-1643 (2002).
[CrossRef]

Zill, D. G.

D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics (PWS-KENT, 1992).

Zou, H.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J.-W. Kang, M.-J. Kim, J.-P. Kim, S.-J. Yoo, J.-S. Lee, D. Y. Kim, and J.-J. Kim, 'Polymeric wavelength filters fabricated using holographic surface relief gratings on azobenzene-containing polymer films,' Appl. Phys. Lett. 82, 3823-3825 (2003).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, 'Coupled-wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

Fiber Integr. Opt. (1)

G. D. Peng and P. L. Chu, 'Polymer optical fiber photosensitivities and highly tunable fiber gratings,' Fiber Integr. Opt. 19, 277-293 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, 'Coupled-mode theory for guided-wave optics,' IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

L. Eldada and L. W. Shacklette, 'Advances in polymer integrated optics,' IEEE J. Sel. Top. Quantum Electron. 6, 54-68 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Tang, Y. Tang, J. Colegrove, and D. M. Craig, 'Fast electro-optic Bragg grating couplers for on-chip reconfigurable optical waveguide interconnects,' IEEE Photon. Technol. Lett. 16, 1385-1387 (2004).
[CrossRef]

Z. Xiong, G. D. Peng, B. Wu, and P. L. Chu, 'Highly tunable Bragg gratings in single-mode polymer optical fibers,' IEEE Photon. Technol. Lett. 11, 352-354 (1999).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, 'Coupled-wave theory of distributed feedback lasers,' J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

J. Lightwave Technol. (2)

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

J. Polym. Sci. Part A: Polym. Chem. (1)

T. Kaino, 'Preparation of plastic optical fibers for near-IR region transmission,' J. Polym. Sci. Part A: Polym. Chem. 25, 37-46 (1987).
[CrossRef]

Opt. Commun. (1)

H. Y. Liu, G. D. Peng, and P. L. Chu, 'Thermal stability of gratings in PMMA and CYTOP polymer fibers,' Opt. Commun. 24, 151-156 (2002).

Opt. Eng. (1)

M. Zhou, 'Low-loss polymeric materials for passive waveguide components in fiber optical communication,' Opt. Eng. 41, 1631-1643 (2002).
[CrossRef]

Opt. Lett. (1)

Other (12)

C.-L. Chen, Elements of Optoelectronics and Fiber Optics (Irwin, 1996).

Product Specification Sheets of Mitsubishi DFB Laser Diodes,http://www.mitsubishichips.com/Global/common/cfm/eProfile.cfm?FOLDER=/product/ opt/laserdiode/optcomld/dfbld (2005).

Product Catalogs of Denselight LEDs,http://www.denselight.com/products%20SLED%20modules%20&%20box%20catalog.htm (2005).

D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics (PWS-KENT, 1992).

K. J. Kim, 'Thermo-structural influences on optical characteristics of polymer Bragg gratings,' Ph.D. dissertation (University of Maryland, 2006).

A. Othonos and K. Kalli, Fiber Bragg Gratings--Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

M. N. Ozisik, Heat Conduction (Wiley, 1980).

A. D. Kraus and A. Bar-Cohen, Thermal Analysis and Control of Electronic Equipment (Hemisphere, 1983).

S. Moaveni, Finite Element Analysis--Theory and Application with ANSYS (Prentice-Hall, 2002).

Product Specification Document (Avensys Inc., 2005).

Product Specification Document (Corning Inc., 2005).

M. J. Weber, Handbook of Laser Science and Technology Supplement 2: Optical Materials (CRC, 1995).

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Figures (11)

Fig. 1
Fig. 1

(Color online) Polymer FBG thermal model: (a) geometry, (b) geometry and∕or boundary conditions for a finite-element model.

Fig. 2
Fig. 2

Iterative solution procedure: fully coupled thermo-optical analysis.

Fig. 3
Fig. 3

(Color online) (a) Heating system for exponentially decaying temperature distribution. (b) Schematic of the measurement system for the spectral analysis.

Fig. 4
Fig. 4

(Color online) Representative excess temperature profiles along the FBG for the excess base temperatures of 85, 65, 45, and 25 K.

Fig. 5
Fig. 5

(Color online) Normalized reflected power spectra for the excess base temperatures of 65 and 25 K.

Fig. 6
Fig. 6

(Color online) Heat generation densities along a PMMA FBG illuminated by SMLDs with spectral bandwidths of (a) 0.001   nm and (b) 0.026   nm at ambient of 25   ° C .

Fig. 7
Fig. 7

(Color online) Analytical and numerical excess temperatures along a PMMA FBG illuminated by SMLDs with spectral bandwidths of (a) 0.001   nm and (b) 0.026   nm at ambient of 25   ° C .

Fig. 8
Fig. 8

(Color online) Thermally induced Bragg wavelength shifts in a PMMA FBG illuminated with SMLDs—thermo-optic (dn∕dT), thermal expansion ( d Λ / d T ) , and combined effects ( d n / d T + d Λ / d T ) associated with (a) 0.001   nm and (b) 0.026   nm of bandwidths of SMLDs at 5 mW of incident optical power with 25   ° C of ambient.

Fig. 9
Fig. 9

(Color online) Reflectivity spectra for a PMMA FBG illuminated by SMLDs with spectral bandwidths of (a) 0.001   nm and (b) 0.026   nm at ambient of 25   ° C .

Fig. 10
Fig. 10

(Color online) Reflected power spectra of a PMMA FBG illuminated by SMLD illuminations with spectral bandwidths of (a) 0.001   nm and (b) 0.026   nm at ambient of 25   ° C .

Fig. 11
Fig. 11

(Color online) (a) Reflectivity-based and (b) reflected-power-based Bragg wavelength shifts in a PMMA FBG illuminated with a 5 mW SMLD with a spectral bandwidth of 0.026   nm at ambient of 25   ° C ; the same information is presented in (a.1)–(b.2) but with different scales of the horizontal axis.

Tables (3)

Tables Icon

Table 1 Properties and Geometry of Fiber Bragg Grating

Tables Icon

Table 2 Predicted and Measured Shifts of Bragg Wavelengths at Various Base Excess Temperatures

Tables Icon

Table 3 Properties and Geometry of PMMA Fiber Bragg Grating

Equations (60)

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

I ( r ) = 2 P i n c π w 2 exp ( 2 r 2 / w 2 ) ,
P ( λ ) ¯ = B exp [ 4 ln 0.5 ( ( λ λ c ) / FWHM ) 2 ] ,
B = 1 2 0 Δ λ c exp [ 4 ln 0.5 ( ( λ λ c ) / FWHM ) 2 ] d ( λ ) ,
λ B = 2 n e f f Λ ,
d R d z = i σ ^ R ( z ) i κ S ( z ) ,
d S d z = i σ ^ S ( z ) + i κ R ( z ) ,
σ ^ = δ a ^ 2 i ,
a ^ = 10 a ln ( 10 ) .
δ = 2 π n e f f λ π Λ .
R ( z ) = r 1 exp ( m z ) + r 2 exp ( m z ) ,
S ( z ) = s 1 exp ( m z ) + s 2 exp ( m z ) ,
m = κ 2 σ ^ 2 = κ 2 ( δ a ^ 2 i ) 2 ,
m = m 1 + i m 2 ,
m 1 = ( ξ 2 + ( a ^ δ ) 2 + ξ ) / 2 ,
m 2 = ( ξ 2 + ( a ^ δ ) 2 ξ ) / 2 ,
ξ = ( a ^ / 2 ) 2 δ 2 + κ 2 .
| R | 2 = c 1 exp ( 2 m 1 z ) + c 2 exp [ 2 m 1 ( 2 L z ) ] + c 3 cos [ 2 m 2 ( z L ) ] + c 4 sin [ 2 m 2 ( z L ) ] c 1 + c 5 + c 2 exp ( 4 m 1 L ) ,
| S | 2 = c 6 { exp ( 2 m 1 z ) + exp [ 2 m 1 ( 2 L z ) ] 2 exp ( 2 L m 1 ) cos [ 2 m 2 ( z L ) ] } c 2 κ 2 exp ( 4 m 1 L ) + c 1 κ 2 + c 7 .
P ( λ , r , z ) = I ( r ) P ( λ ) ¯ ( | R ( λ , z ) | 2 + | S ( λ , z ) | 2 ) .
q G ( λ , r , z ) = P ( λ , r , z ) a ^ ( λ , r , z ) P ( λ , r , z ) a ^ .
q G ( r , z ) = q G ( λ , r , z ) d λ = a ^ I ( r ) P ( λ ) ¯ ( | R ( λ , z ) | 2 + | S ( λ , z ) | 2 ) d λ .
q G ( z ) = P i n c π r o 2 P ( λ ) ¯ ( | R ( λ ,   z ) | 2 + | S ( λ ,   z ) | 2 ) d λ a ^ ,
d 2 θ d z 2 p 2 θ + 1 k P i n c π r o 2 P ( λ ) ¯ ( | R ( λ , z ) | 2 + | S ( λ , z ) | 2 ) d λ a ^ = 0 ,
h = 1 r o k ln r c l r o + r o h c l r c l r c l r o h c l ,
θ = d 1 exp ( p z ) + d 2 exp ( p z ) + θ p ,
θ p = { F ( λ ) exp [ 2 m 1 ( λ ) z ] + G ( λ ) exp [ 2 m 1 ( λ ) z ] + H ( λ ) cos ( M ( λ ) ( z L ) ) + N ( λ ) sin ( U ( λ ) ( z L ) ) } d λ .
| d θ d z | z = 0 = 0 , | d θ d z | z = L = 0 .
[ R M ( λ ) S M ( λ ) ] = F ( λ ) [ R ( L ) S ( L ) ] ;
F ( λ ) = F M ( λ ) F M 1 ( λ ) F j ( λ ) F 1 ( λ ) ,
F j ( λ ) = [ cosh ( m j ( λ ) Δ z ) i σ ^ j ( λ ) m j ( λ ) sinh ( m j ( λ ) Δ z ) i κ ( λ ) m j ( λ ) sinh ( m j ( λ ) Δ z ) i κ ( λ ) m j ( λ ) sinh ( m j ( λ ) Δ z ) cosh ( m j ( λ ) Δ z ) + i σ ^ j ( λ ) m j ( λ ) sinh ( m j ( λ ) Δ z ) ] ,
Δ n e f f = d n d T Δ T .
Δ Λ = Λ α Δ T ,
n e f f 2 , j = n e f f 1 + d n d T Δ T j .
Λ 2 , j = Λ 1 ( 1 + α Δ T j ) .
σ ^ j ( λ ) = 2 π n e f f j λ π Λ j i a ^ 2 = 2 π λ ( n e f f 1 + d n d T Δ T j ) π Λ 1 ( 1 + α Δ T j ) i a ^ 2 ,
m j ( λ ) = κ 2 ( λ ) σ ^ j 2 ( λ ) = ( π λ Δ n e f f ) 2 [ 2 π λ ( n e f f 1 + d n d T Δ T j ) π Λ 1 ( 1 + α Δ T j ) i a ^ 2 ] 2 .
ρ ( λ ) = | S M ( λ ) | 2 | R M ( λ ) | 2 .
P r e f ( λ ) = P i ( λ ) ρ ( λ ) ,
c 1 = ( m 1 a ^ / 2 ) 2 + ( m 2 δ ) 2 ,
c 2 = ( m 1 + a ^ / 2 ) 2 + ( m 2 + δ ) 2 ,
c 3 = 2 exp ( 2 m 1 L ) ( m 1 2 + m 2 2 ( a ^ / 2 ) 2 δ 2 ) ,
c 4 = 4 exp ( 2 m 1 L ) ( m 1 δ + m 2 a ^ / 2 ) ,
c 5 = c 3 cos ( 2 m 2 L ) c 4 sin ( 2 m 2 L ) ,
c 6 = ( m 1 2 m 2 2 ( a ^ / 2 ) 2 + δ 2 ) 2 + 4 ( m 1 m 2 a ^ / 2 δ ) 2 ,
c 7 = κ 2 c 5 .
F ( λ ) = P i n c a ^ P ( λ ) ¯ a 1 ( λ ) / f 3 ,
G ( λ ) = P i n c a ^ P ( λ ) ¯ a 2 ( λ ) exp [ 4 L m 1 ( λ ) ] / f 3 ,
M ( λ ) = U ( λ ) = 2 m 2 ( λ ) ,
H ( λ ) = P i n c a ^ P ( λ ) ¯ a 3 ( λ ) / f 4 ,
N ( λ ) = P i n c a ^ P ( λ ) ¯ a 4 ( λ ) / f 4 ,
a 1 ( λ ) = c 1 / f 1 + f 2 ,
a 2 ( λ ) = c 2 / f 1 + f 2 ,
a 3 ( λ ) = c 3 / f 1 2 exp ( 2 m 1 L ) f 2 ,
a 4 ( λ ) = c 4 / f 1 ,
f 1 = c 1 + c 5 + c 2 exp ( 4 m 1 L ) ,
f 2 = c 6 / ( κ 2 c 2 exp ( 4 m 1 L ) + 2 κ 2 c 5 + κ 2 c 1 ) ,
f 3 = k π r o 2 ( 4 ( m 1 ( λ ) ) 2 p 2 ) ,
f 4 = k π r o 2 ( 4 ( m 2 ( λ ) ) 2 + p 2 ) .
  d 1 = 1 p ( 1 exp ( 2 p L ) ) × [ { 2 m 1 ( λ ) [ F ( λ ) ( 1 exp [ L ( 2 m 1 ( λ ) + p ) ] ) G ( λ ) ( 1 exp [ L ( 2 m 1 ( λ ) p ) ] ) ] H ( λ ) M ( λ ) sin ( L M ( λ ) ) + N ( λ ) M ( λ ) ( cos ( L M ( λ ) ) exp ( p L ) ) } d λ ] ,
d 2 = d 1 + 1 p [ { 2 m 1 ( λ ) ( F ( λ ) G ( λ ) ) + H ( λ ) M ( λ ) × sin ( L M ( λ ) ) + N ( λ ) M ( λ ) cos ( L M ( λ ) ) } d λ ] .

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