Abstract

Analysis of previous measurements of thermal phase sensitivity in hollow core photonic crystal fibers is presented with additional new corroborating measurements, resolving a discrepancy in previously reported results. We extend an existing derivation of thermo-mechanical phase sensitivity in solid- and hollow-core photonic crystal fiber to also include kagome lattice photonic crystal fibers. Measured thermal phase response is shown to agree with theoretical prediction to within a few percent.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (3)

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

D. S. Sanditov and M. V. Darmaev, “Elactic moduli and poisson’s coeficient of optical glasses,” Russian Physics Journal 58, 683–690 (2015).
[Crossref]

G. A. Cranch and G. A. Miller, “Coherent light transmission properties of comercial photonic crystal hollow core optical fiber,” Appl. Opt. 54, F8–F16 (2015).
[Crossref] [PubMed]

2014 (3)

2013 (1)

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (1)

V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13, 6679–6684 (2005).
[Crossref]

2004 (1)

A.-J. Wang and D. L. McDowell, “In-plane stiffness and yield strength of periodic metal honeycombs,” Journal of Engineering Materials and Technology 126, 137–156 (2004).
[Crossref]

2000 (1)

B. Kim and R. M. Christensen, “Basic two-dimension core types for sandwich structures,” International Journal of Mechanical Sciences 42, 657–676 (2000).
[Crossref]

1999 (2)

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

S. W. Løvseth, J. T. Kringlebotn, E. Rønnekliev, and K. Bløtekær, “Fiber distributed-feedback lasers used as acoustic sensors in air,” Appl. Opt. 38, 4821–4830 (1999).
[Crossref]

1997 (1)

H. Lefévre, “Fundamentals of the interferometric fiber-optic gyroscope,” Optical Review 4, 20–27 (1997).
[Crossref]

1983 (1)

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

1981 (1)

1980 (2)

1979 (3)

1978 (1)

A. H. Hartog, A. J. Conduit, and D. N. Payne, “Variation of pulse delay with stress and temperature in jacketed and unjacketed optical fibres,” Optical and Quantum Electronics 11, 265–273 (1978).
[Crossref]

1963 (1)

R. E. Barker., “An approximate relation between elastic moduli and thermal expansivities,” J. Appl. Phys. 34, 107–116 (1963).
[Crossref]

Allan, D.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Argyros, A.

Ashby, M. F.

L. J. Gibson and M. F. Ashby, Cellular Solids: Structures and Properties, 2nd ed. (Cambridge University, 1997).
[Crossref]

Baddela, N.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Barker, R. E.

R. E. Barker., “An approximate relation between elastic moduli and thermal expansivities,” J. Appl. Phys. 34, 107–116 (1963).
[Crossref]

Birks, T.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Blin, S.

Bløtekær, K.

Bucaro, J. A.

N. Lagakos, J. A. Bucaro, and J. Jarzynski, “Termperature-induced optical phase shifts in fibers,” Appl. Opt. 20, 2305–2308 (1981).
[Crossref] [PubMed]

J. A. Bucaro, N Lagakos, J. H. Cole, and T. G. Giallorenzi, “Fiber optic acoustic transduction,” in Physical Acoustics Vol. XVI: Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, 1982).
[Crossref]

Budiansky, B.

Cao, Y.

Cassidy, S. A.

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

Christensen, R. M.

B. Kim and R. M. Christensen, “Basic two-dimension core types for sandwich structures,” International Journal of Mechanical Sciences 42, 657–676 (2000).
[Crossref]

Cole, J. H.

J. A. Bucaro, N Lagakos, J. H. Cole, and T. G. Giallorenzi, “Fiber optic acoustic transduction,” in Physical Acoustics Vol. XVI: Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, 1982).
[Crossref]

Conduit, A. J.

A. H. Hartog, A. J. Conduit, and D. N. Payne, “Variation of pulse delay with stress and temperature in jacketed and unjacketed optical fibres,” Optical and Quantum Electronics 11, 265–273 (1978).
[Crossref]

Cranch, G. A.

Cregan, R.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Dangui, V.

V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13, 6679–6684 (2005).
[Crossref]

Darmaev, M. V.

D. S. Sanditov and M. V. Darmaev, “Elactic moduli and poisson’s coeficient of optical glasses,” Russian Physics Journal 58, 683–690 (2015).
[Crossref]

Digonnet, M. J. F.

S. Blin, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Reduced thermal sensitivity of a fiber-optic gyroscope using an air-core photonic-bandgap fiber,” J. Lightwave Technol. 25, 861–865 (2007).
[Crossref]

V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13, 6679–6684 (2005).
[Crossref]

Docherty, A.

Drucker, D. C.

Fokoua, E. Numkam

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Giallorenzi, T. G.

J. A. Bucaro, N Lagakos, J. H. Cole, and T. G. Giallorenzi, “Fiber optic acoustic transduction,” in Physical Acoustics Vol. XVI: Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, 1982).
[Crossref]

Gibson, L. J.

L. J. Gibson and M. F. Ashby, Cellular Solids: Structures and Properties, 2nd ed. (Cambridge University, 1997).
[Crossref]

Gray, D. R.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Hartog, A. H.

A. H. Hartog, A. J. Conduit, and D. N. Payne, “Variation of pulse delay with stress and temperature in jacketed and unjacketed optical fibres,” Optical and Quantum Electronics 11, 265–273 (1978).
[Crossref]

Hayes, J. R.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Ho, H. L.

Hocker, G. B.

Hornung, S.

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

Hughes, R.

Jarzynski, J.

Jin, W.

Ju, J.

Kashyap, R.

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

Kim, B.

B. Kim and R. M. Christensen, “Basic two-dimension core types for sandwich structures,” International Journal of Mechanical Sciences 42, 657–676 (2000).
[Crossref]

Kim, H. K.

S. Blin, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Reduced thermal sensitivity of a fiber-optic gyroscope using an air-core photonic-bandgap fiber,” J. Lightwave Technol. 25, 861–865 (2007).
[Crossref]

V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13, 6679–6684 (2005).
[Crossref]

Kino, G. S.

Knight, J.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Kringlebotn, J. T.

Lagakos, N

J. A. Bucaro, N Lagakos, J. H. Cole, and T. G. Giallorenzi, “Fiber optic acoustic transduction,” in Physical Acoustics Vol. XVI: Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, 1982).
[Crossref]

Lagakos, N.

Lefévre, H.

H. Lefévre, “Fundamentals of the interferometric fiber-optic gyroscope,” Optical Review 4, 20–27 (1997).
[Crossref]

Leon-Saval, S. G.

Li, Z.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Løvseth, S. W.

Mangan, B.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Marra, G.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

McDowell, D. L.

A.-J. Wang and D. L. McDowell, “In-plane stiffness and yield strength of periodic metal honeycombs,” Journal of Engineering Materials and Technology 126, 137–156 (2004).
[Crossref]

Miller, G. A.

Numkam Fokoua, E.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

Pang, M.

Payne, D. N.

A. H. Hartog, A. J. Conduit, and D. N. Payne, “Variation of pulse delay with stress and temperature in jacketed and unjacketed optical fibres,” Optical and Quantum Electronics 11, 265–273 (1978).
[Crossref]

Petrovich, M.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

Petrovich, M. N.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Pla, J.

Poletti, F.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22, 23807–23828 (2014).
[Crossref] [PubMed]

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Reeve, M. H.

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

Rice, J. R.

Richardson, D. J.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Roberts, P.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Rønnekliev, E.

Russell, P.

R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Sanditov, D. S.

D. S. Sanditov and M. V. Darmaev, “Elactic moduli and poisson’s coeficient of optical glasses,” Russian Physics Journal 58, 683–690 (2015).
[Crossref]

D. S. Sanditov and B. S. Sydykon, “Modulus of elasticity and thermal expansion coefficient of glassy solids,” Physics of the Solid State 56, 1006–1008 (2014).
[Crossref]

Slavik, R.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Sugawara, Y.

Sydykon, B. S.

D. S. Sanditov and B. S. Sydykon, “Modulus of elasticity and thermal expansion coefficient of glassy solids,” Physics of the Solid State 56, 1006–1008 (2014).
[Crossref]

Tanaka, S.

Tateda, M.

Tinder, R. F.

R. F. Tinder, Tensor Properties of Solids (Morgan & Claypool, 2008).

Wang, A.-J.

A.-J. Wang and D. L. McDowell, “In-plane stiffness and yield strength of periodic metal honeycombs,” Journal of Engineering Materials and Technology 126, 137–156 (2004).
[Crossref]

Wheeler, N. V.

R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015).
[Crossref] [PubMed]

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013).
[Crossref]

Xuan, H. F.

Yang, F.

Appl. Opt. (8)

Electron. Lett. (1)

R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983).
[Crossref]

International Journal of Mechanical Sciences (1)

B. Kim and R. M. Christensen, “Basic two-dimension core types for sandwich structures,” International Journal of Mechanical Sciences 42, 657–676 (2000).
[Crossref]

J. Appl. Phys. (1)

R. E. Barker., “An approximate relation between elastic moduli and thermal expansivities,” J. Appl. Phys. 34, 107–116 (1963).
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[Crossref]

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NKT Photonics Inc., Technical Specifications for HC19-1550, 1400 Campus Drive West, Morganville, NJ 07751, USA (2016).

NKT Photonics Inc., Technical Specifications for HC-1550-02, 1400 Campus Drive West, Morganville, NJ 07751, USA (2016).

GLOphotonics, Technical Specifications for PMC-C-TiSa_Er-7C, 1 Avenue d’Ester, 87069 Limoges Cedex, France (2016).

Corning, Technical Specifications for SMF-28, One Riverfront Plaza, Corning, NY 14831 USA (2016).

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Figures (5)

Fig. 1
Fig. 1 Experimental diagram for measuring the thermal response of hollow-core fibers, and SMF-28.
Fig. 2
Fig. 2 Examples of fiber thermal phase response (blue) and corresponding temperature (orange) measurements. Clockwise from top left: large core photonic bandgap fiber, HC19-1550; large core inhibited coupling kagome type fiber, PMC-C-TiSa_Er-7C; solid core, SMF-28; small core photonic bandgap fiber, HC-1550-02.
Fig. 3
Fig. 3 Transverse cross-section of hollow-core photonic bandgap fiber. The hollow core extends from the origin to r = a, an inner silica core-sheath from arb, the honeycomb of the photonic crystal structure brc, and finally the outer silica cladding ≤ crd.
Fig. 4
Fig. 4 A regular hexagonal honeycomb crystalline structure. This photonic crystal is made from close-packed regular hexagons whose side length is l and whose side thickness is t. The hexagon edges in this case represent the silica material.
Fig. 5
Fig. 5 A hexagonal kagome crystalline structure. This photonic crystal is made from equilateral triangles and regular hexagons whose side length are l and whose side thickness are t.

Tables (3)

Tables Icon

Table 1 Thermal phase response and time delay change for fibers presented in [7, 8]. Numbers in parentheses are derived from Eq. 6, assuming n = 1 for hollow core and n = 1.45 for solid core fibers, yielding scaling coefficient values of ∼ 3.33 and ~ 4.83, respectively. Reported length is in meters. Units for Sϕ are in ppm/K, whereas units for (1/L)dτ/dT are given in ps/km/K.

Tables Icon

Table 2 Experimental thermal phase response, and numerical modeling of the response due to fiber elongation for the three hollow core fibers. Numbers in parentheses are derived from Eq. 6, assuming n = 1 for hollow core and n = 1.458 for solid core fibers, yielding scaling coefficient values of ∼ 3.33 and ∼ 4.83, respectively. Length is in meters. Units for Sϕ and SL are in ppm/K, and units for (1/L) /dT are in ps/km/K.

Tables Icon

Table 3 Elastic properties of two crystal-like structures and fused silica in terms of the corresponding property in fused silica, see [18–20]. The air filling ratio η relates to the cell wall thickness, t, and wall length, l, by η = ( 1 2 t / 3 l ). Crystal structures are as shown in Fig. 45.

Equations (32)

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τ = n g L c ,
1 L d τ d T = 1 c L ( L d n g d T + n g d L d T ) .
ϕ = k n L .
S ϕ 1 ϕ d ϕ d T = 1 n d n d T + 1 L d L d T , = S n + S L
D τ 1 τ d τ d T = 1 n d n d T + 1 L d L d T , = D n + D L
1 L d τ d T = ( n c ) S ϕ .
S ϕ 1 ϕ ( Δ ϕ Δ t ) ( Δ T Δ t ) 1
S ϕ S L = ϵ z Δ T .
[ ϵ 11 ϵ 22 ϵ 33 ] = [ 1 E 1 ν 21 E 2 ν 31 E 3 ν 12 E 1 1 E 2 ν 32 E 3 ν 13 E 1 ν 23 E 2 1 E 3 ] [ σ 11 σ 22 σ 33 ] ,
[ ϵ r ϵ θ ϵ z ] = [ 1 E S ν S E S ν S E S ν S E S 1 E S ν S E S ν S E S ν S E S 1 E S ] [ σ r σ θ σ z ] .
[ σ r σ θ σ z ] = [ ( λ + 2 μ ) λ λ λ ( λ + 2 μ ) λ λ λ ( λ + 2 μ ) ] [ ϵ r ϵ θ ϵ z ] ,
[ ϵ r ϵ θ ϵ z ] = [ 1 E T h e x 1 E T h e x ν S E L h e x 1 E T h e x 1 E T h e x ν S E L h e x ν S E L h e x ν S E L h e x 1 E L h e x ] [ σ r σ θ σ z ] .
[ σ r σ θ σ z ] = ψ h e x [ β h e x γ h e x δ h e x γ h e x β h e x δ h e x δ h e x δ h e x 0 ] [ ϵ r ϵ θ ϵ z ] ,
[ ϵ r ϵ θ ϵ z ] = [ 1 E T k a g 1 3 E T k a g ν S E L k a g 1 3 E T k a g 1 E T k a g ν S E L k a g ν S E L k a g ν S E L k a g 1 E L k a g ] [ σ r σ θ σ z ] .
[ σ r σ θ σ z ] = ψ k a g [ β k a g γ k a g δ k a g γ k a g β k a g δ k a g δ k a g δ k a g 1 ] [ ϵ r ϵ θ ϵ z ] ,
[ ϵ r ϵ θ ϵ z ] = [ ϵ r ϵ θ ϵ z ] [ α r α θ α z ] Δ T
ϵ r , i = K 1 , i + K 2 , i r 2 ,
ϵ θ , i = K 1 , i K 2 , i r 2 ,
ϵ z , i = K 3 , i .
ϵ z , i = ϵ z , 2 = ϵ z , 3 = K 3 ,
ϵ z , 1 K 3 , 1 α z , 1 Δ T ,
ϵ z , 2 K 3 , 2 α z , 2 Δ T ,
ϵ z , 3 K 3 , 3 α z , 3 Δ T ,
ϵ z , 1 = ϵ z , 2 = ϵ z , 3 .
F z = 0 = i σ z , i A i 0 2 π 0 R σ z r d θ d r
σ r , 1 | r = a = P 1 ,
σ r , 1 | r = b = σ r , 2 | r = b ,
σ r , 2 | r = c = σ r , 3 | r = c ,
σ r , 3 | r = d = P 2 ,
u r , i = ϵ r , i d r
u r , 1 | r = b = u r , 2 | r = b ,
u r , 2 | r = c = u r , 3 | r = c .

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