Abstract

We present a theoretical study of transverse mode instability (TMI) in non-circular ytterbium-doped fibers including the rectangular core in a circular or D-shaped cladding. The D-shaped cladding is found efficient to suppress the TMI thanks to better heat dissipation, as compared to the circular cladding. However, the rectangular core does not suppress the TMI despite its better heat dissipation than a circular core counterpart. Although the temperature built in the rectangular core decreases with an increasing aspect ratio of the rectangular core, the low temperature does not benefit the TMI suppression. Instead, the TMI becomes stronger than its circular core counterpart. Our study reveals that the power coupling between two involved modes and gain saturation effect play a significant role in influencing the TMI. The power coupling strength is associated with the frequency offset between two modes, and it grows with an increasing aspect ratio of rectangular cores, suggesting the longer axis of rectangular core promotes the TMI.

© 2017 Optical Society of America

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References

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2016 (3)

2014 (2)

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 472–483 (2014).
[Crossref]

D. R. Drachenberg, M. J. Messerly, P. H. Pax, A. K. Sridharan, J. B. Tassano, and J. W. Dawson, “Yb3+ doped ribbon fiber for high-average power lasers and amplifiers,” Proc. SPIE 8961, 89610T (2014).
[Crossref]

2013 (9)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fiber lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21(3), 2606–2623 (2013).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
[Crossref] [PubMed]

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21(10), 12053–12067 (2013).
[Crossref] [PubMed]

W. W. Ke, X. J. Wang, X. F. Bao, and X. J. Shu, “Thermally induced mode distortion and its limit to power scaling of fiber lasers,” Opt. Express 21(12), 14272–14281 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013).
[Crossref] [PubMed]

S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013).
[Crossref] [PubMed]

D. Drachenberg, M. Messerly, P. Pax, A. Sridharan, J. Tassano, and J. Dawson, “First multi-watt ribbon fiber oscillator in a high order mode,” Opt. Express 21(15), 18089–18096 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (4)

2010 (1)

2008 (1)

2006 (1)

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Transfer 128(7), 709–716 (2006).
[Crossref]

2001 (1)

K. D. Cole and D. H. Yen, “Green’s functions, temperature and heat flux in the rectangle,” J. Heat Transfer 44(20), 3883–3894 (2001).
[Crossref]

1980 (2)

1978 (1)

Alkeskjold, T. T.

Armstrong, J. P.

Bao, X. F.

Barty, C. P. J.

Beach, R. J.

Broeng, J.

Clarkson, W. A.

Cole, K. D.

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Transfer 128(7), 709–716 (2006).
[Crossref]

K. D. Cole and D. H. Yen, “Green’s functions, temperature and heat flux in the rectangle,” J. Heat Transfer 44(20), 3883–3894 (2001).
[Crossref]

Dajani, I.

Dawson, J.

Dawson, J. W.

Dong, L.

Drachenberg, D.

Drachenberg, D. R.

D. R. Drachenberg, M. J. Messerly, P. H. Pax, A. K. Sridharan, J. B. Tassano, and J. W. Dawson, “Yb3+ doped ribbon fiber for high-average power lasers and amplifiers,” Proc. SPIE 8961, 89610T (2014).
[Crossref]

A. K. Sridharan, P. H. Pax, J. E. Heebner, D. R. Drachenberg, J. P. Armstrong, and J. W. Dawson, “Mode-converters for rectangular-core fiber amplifiers to achieve diffraction-limited power scaling,” Opt. Express 20(27), 28792–28800 (2012).
[Crossref] [PubMed]

Eidam, T.

Feit, M. D.

Fleck, J. A.

Hansen, K. R.

Heebner, J. E.

Ho, D.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Jansen, F.

Jauregui, C.

Ji, J.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Kalra, Y.

Ke, W. W.

Kumar, A.

Lægsgaard, J.

Limpert, J.

Madden, T.

Marciante, J. R.

Messerly, M.

Messerly, M. J.

Naderi, S.

Nilsson, J.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Otto, H. J.

Pax, P.

Pax, P. H.

Raghurman, S.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Richardson, D. J.

Robin, C.

Rockwell, D. A.

Sahu, J.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Saini, T. S.

Schmidt, O.

Schreiber, T.

Shkunov, V. V.

Shu, X. J.

Shverdin, M. Y.

Siders, C. W.

Sinha, R. K.

Smith, A. V.

Smith, J. J.

Sridharan, A.

Sridharan, A. K.

Stappaerts, E. A.

Stutzki, F.

Tassano, J.

Tassano, J. B.

D. R. Drachenberg, M. J. Messerly, P. H. Pax, A. K. Sridharan, J. B. Tassano, and J. W. Dawson, “Yb3+ doped ribbon fiber for high-average power lasers and amplifiers,” Proc. SPIE 8961, 89610T (2014).
[Crossref]

Tunnermann, A.

H. J. Otto, C. Jauregui, J. Limpert, and A. Tunnermann, “Average power limit of fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

Tünnermann, A.

Wang, X. J.

Ward, B.

Ward, B. G.

Wirth, C.

Wu, X.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Xia, N.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Yen, D. H.

K. D. Cole and D. H. Yen, “Green’s functions, temperature and heat flux in the rectangle,” J. Heat Transfer 44(20), 3883–3894 (2001).
[Crossref]

Yoo, S.

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

Appl. Opt. (4)

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

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 472–483 (2014).
[Crossref]

J. Heat Transfer (2)

K. D. Cole and D. H. Yen, “Green’s functions, temperature and heat flux in the rectangle,” J. Heat Transfer 44(20), 3883–3894 (2001).
[Crossref]

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Transfer 128(7), 709–716 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fiber lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Opt. Express (16)

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

D. A. Rockwell, V. V. Shkunov, and J. R. Marciante, “Semi-guiding high-aspect-ratio core (SHARC) fiber providing single-mode operation and an ultra-large core area in a compact coilable package,” Opt. Express 19(15), 14746–14762 (2011).
[Crossref] [PubMed]

D. Drachenberg, M. Messerly, P. Pax, A. Sridharan, J. Tassano, and J. Dawson, “First multi-watt ribbon fiber oscillator in a high order mode,” Opt. Express 21(15), 18089–18096 (2013).
[Crossref] [PubMed]

A. K. Sridharan, P. H. Pax, J. E. Heebner, D. R. Drachenberg, J. P. Armstrong, and J. W. Dawson, “Mode-converters for rectangular-core fiber amplifiers to achieve diffraction-limited power scaling,” Opt. Express 20(27), 28792–28800 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21(3), 2606–2623 (2013).
[Crossref] [PubMed]

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21(10), 12053–12067 (2013).
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
[Crossref] [PubMed]

S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
[Crossref] [PubMed]

W. W. Ke, X. J. Wang, X. F. Bao, and X. J. Shu, “Thermally induced mode distortion and its limit to power scaling of fiber lasers,” Opt. Express 21(12), 14272–14281 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013).
[Crossref] [PubMed]

L. Dong, “Thermal lensing in optical fibers,” Opt. Express 24(17), 19841–19852 (2016).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (2)

D. R. Drachenberg, M. J. Messerly, P. H. Pax, A. K. Sridharan, J. B. Tassano, and J. W. Dawson, “Yb3+ doped ribbon fiber for high-average power lasers and amplifiers,” Proc. SPIE 8961, 89610T (2014).
[Crossref]

H. J. Otto, C. Jauregui, J. Limpert, and A. Tunnermann, “Average power limit of fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016).

Other (2)

S. Yoo, J. Ji, X. Wu, S. Raghurman, D. Ho, N. Xia, J. Sahu, and J. Nilsson, “Mode area scalability in rectangular core fiber,” in Proceedings of IEEE Conference on Photonics (IEEE, 2015), pp. 323–324.
[Crossref]

A. V. Smith and J. J. Smith, “Frequency dependence of mode coupling gain in Yb doped fiber amplifiers due to stimulated thermal Rayleigh scattering,” arXiv: 1301.4277 [physics. optics] (2013).

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Figures (20)

Fig. 1
Fig. 1 Intensity of FM (LP01) and HOM (LP11) in (a) 50 μm circular core (Fiber 1), (b) square core 44.3 μm × 44.3 μm (AR. 1:1) (Fiber 2), (c) rectangular core 88.8 μm × 22.2 μm (AR. 4:1) (Fiber 3) and (d) rectangular core 140 μm × 14 μm (AR. 10:1) (Fiber 4). The dashed lines indicate the thermal boundary.
Fig. 2
Fig. 2 Normalized output LP11 mode power content versus frequency offset for 50 μm circular core fiber (Fiber 1), 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), 88.8 μm × 22.2 μm rectangular core (AR. 4:1) fiber (Fiber 3) and 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4).
Fig. 3
Fig. 3 Increased temperature distribution (at t = 0) at the launch end in (a) 50 μm circular core fiber (Fiber 1), (b) 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), (c) 88.8 μm × 22.2 μm rectangular core (AR. 4:1) fiber (Fiber 3) and (d) 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4).
Fig. 4
Fig. 4 Increased temperature profile (at t = 0) evolution in the core along fiber length for (a) 50 μm circular core fiber (Fiber 1), (b) 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), (c) 88.8 μm × 22.2 μm rectangular core (AR. 4:1) fiber (Fiber 3) and (d) 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4). The same launched pump and signal powers are assumed as in Fig. 3.
Fig. 5
Fig. 5 Time averaged powers versus fiber length for (a) 50 μm circular core fiber (Fiber 1), (b) 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), (c) 88.8 μm × 22.2 μm rectangular core (AR. 4:1) fiber (Fiber 3) and (d) 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4). The same launched pump and signal powers are assumed as in Fig. 3.
Fig. 6
Fig. 6 The output LP11 power content versus the input pump power for 50 μm circular core fiber (Fiber 1), 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), 88.8 μm × 22.2 μm rectangular core (AR. 4:1) fiber (Fiber 3) and 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4). The same launched signal powers are assumed as in Fig. 3.
Fig. 7
Fig. 7 The normalized signal intensity and upper state population (n2) profiles at the input end for (a) 50 μm circular core fiber (Fiber 1), (b) 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), (c) 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4). Insets enlarge the selected portion to emphasize the overlap differences by adjusting pump intensities. The same launched pump and signal powers are assumed as in Fig. 3. PR represents the pumping radius.
Fig. 8
Fig. 8 The output LP11 power content versus the input pump power for 50 μm circular core fiber (Fiber 1), 44.3 μm × 44.3 μm square core (AR. 1:1) fiber (Fiber 2), and 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4). The same launched pump and signal powers are assumed as in Fig. 3.
Fig. 9
Fig. 9 The signal effective area as a function of propagation length, z, for (a) 50 μm circular core fiber (Fiber 1) and (b) 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) when the same launched pump and signal powers are assumed as in Fig. 3.
Fig. 10
Fig. 10 The threshold pump power versus core area for circular core, square core (AR. 1:1), rectangular core (AR. 4:1) and rectangular core (AR. 10:1) fibers. The same launched signal powers are assumed as in Fig. 3.
Fig. 11
Fig. 11 The threshold pump power (left axis) and power coupling coefficient χ (right axis) versus effective mode area for circular core, rectangular core (AR. 1:1), rectangular core (AR. 4:1) and rectangular core (AR. 10:1) fibers. The same launched signal powers are assumed as in Fig. 3.
Fig. 12
Fig. 12 Intensity of (a) LP01 and (b) LP11 mode in the 50 μm circular core fiber (Fiber 1) with D-shaped cladding, (c) LP01 and (d) LP11 in the 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) with D-shaped cladding. The same core parameters are used as listed in Table 1. The dashed lines indicate the thermal boundary position.
Fig. 13
Fig. 13 Increased temperature distribution (at t = 0) within the whole core and cladding region at the launch end in 50 μm circular core (Fiber 1) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers are assumed as in Fig. 3. The input pump power is 250 W with 100 μm pumping radius.
Fig. 14
Fig. 14 Increased temperature profile evolution (at t = 0) within the core along fiber in 50 μm circular core (Fiber 1) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers, pump power and pumping radius are assumed as in Fig. 13.
Fig. 15
Fig. 15 Time averaged powers versus fiber length for 50 μm circular core (Fiber 1) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers, pump power and pumping radius are assumed as in Fig. 13.
Fig. 16
Fig. 16 The output LP11 power content (left axis) and maximum increased temperature in the core (right axis) versus the input pump power for 50 μm circular core (Fiber 1) with circular cladding and D-shaped cladding. The same launched signal powers and pumping radius are assumed as in Fig. 13.
Fig. 17
Fig. 17 Increased temperature distribution (at t = 0) at the launch end within the whole core and cladding region in 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers are assumed as in Fig. 3. The input pump power is 140 W with 100 μm pumping radius.
Fig. 18
Fig. 18 Increased temperature profile evolution (at t = 0) in the core along fiber for 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers and pumping radius are assumed as in Fig. 17.
Fig. 19
Fig. 19 Time averaged powers versus fiber length for 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) with (a) circular cladding and (b) D-shaped cladding. The same launched signal powers and pumping radius are assumed as in Fig. 17.
Fig. 20
Fig. 20 The output LP11 power content (left axis) and maximum increased temperature in the core (right axis) versus the input pump power for 140 μm × 14 μm rectangular core (AR. 10:1) fiber (Fiber 4) with circular cladding and D-shaped cladding. The same launched signal powers and pumping radius are assumed as in Fig. 17.

Tables (2)

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Table 1 Fiber parameters used in simulation

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Table 2 Output powers in the investigated fibers

Equations (24)

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E s (x,y,t)= P 01 E 01 (x,y)+ P 11 E 11 (x,y) e iΔωt
E s (x,y,z,t) z = i 2 k c 2 E s (x,y,z,t) i[ k 2 (x,y,z,t) k c 2 ] 2 k c E s (x,y,z,t)+g(x,y,z,t) E s (x,y,z,t)
k(x,y,z,t)= ω c n core (x,y,z,t) c
g(x,y,z,t)= 1 2 [ σ s a +( σ s a + σ s e ) n u (x,y,z,t)] N Yb (x,y)
n core (x,y,z,t)= n core + dn dT ΔT(x,y,z,t)
n u (x,y,z,t)= P p (z,t) σ p a h ν p A p + I s (x,y,z,t) σ s a h ν s P p (z,t)( σ p a + σ p e ) h ν p A p + I s (x,y,z,t)( σ s a + σ s e ) h ν s + 1 τ
E s (x,y,z,t) z = i 2 k c 2 E s (x,y,z,t)
E s (x,y,z,t)= 1 2π E s ( k x , k y ,z,t) e i k x x e i k y y d k x d k y
E s ( k x , k y ,z,t) z =i k x 2 2 k c E s ( k x , k y ,z,t)+i k y 2 2 k c E s ( k x , k y ,z,t)
ϕ( k x , k y )= Δz 2 ( k x 2 + k y 2 ) 2 k c
E s (x,y,z,t) z = i[ k 2 (x,y,z,t) k c 2 ] 2 k c E s (x,y,z,t)+g(x,y,z,t) E s (x,y,z,t)
ϕ(x,y,t)=Δz k 2 (x,y,t) k c 2 2 k c
G(x,y,ω| x',y')= n=0 F n (y,y') P n (x,x', ω)
F n (y,y')= 1 WK sin( nπ W y)sin( nπ W y')
P n (x,x',ω)={ exp[ σ n (2H| xx' | )]exp[ σ n (2Hxx')] +exp[ σ n | xx' |] exp[ σ n (x+x')] }/ σ n (1exp[2 σ n H])
σ n 2 = ( nπ W ) 2 +iω ρC K
ρC ΔT(x,y,z,t) t =Q(x,y,z,t)+K( 2 ΔT(x,y,z,t) x 2 + 2 ΔT(x,y,z,t) y 2 )
Q(x,y,z,t)= N Yb (x,y)[ ν p ν s ν p ][ σ p a ( σ p a + σ p e ) n u (x,y,z,t)] P p (z,t) A p
ΔT(x,y,t)=Real[ m=0 1 x',y' q(x',y', ω m ) G(x,y, ω m | x',y' ) exp(i ω m t)]
q(x',y', ω m )=ΔxΔy i=0 N t 1 Q(x',y', t i )exp(i ω m t i )
d P p (z,t) dz = P p (z,t) A p [( σ p a + σ p e ) n u (x,y,z,t) σ p a ] N Yb (x,y)dxdy
P 01 (z,t)= | E s (x,y,z,t) E 01 (x,y)dxdy [ E 01 (x,y)] 2 dxdy | 2 P 11 (z,t)= | E s (x,y,z,t) E 11 (x,y)dxdy [ E 11 (x,y)] 2 dxdy | 2
χ 1,2 (Δω)= dn dT k 2 K β 1,2 Im[A(Δω)](1 λ s λ p )
Im(A(Δω))= Ω E 01 (x,y) E 11 (x,y)× ( Ω d Im(G(x,y,x',y',Δω)) E 01 (x',y') E 11 (x',y') dx'dy')dxdy

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