Abstract

Silicon platform enables the monolithic realization of large-scale photonic integrated systems. Many emerging applications facilitated by silicon photonics such as optical biosensing, optical neurostimulation, optical phased arrays, holographic displays, 3D cameras, optical machine learning, and optical quantum information processing systems require the integration of a large number of optical phase modulators with modest modulation speed. Classical optical modulators are not suitable for such large-scale integration because of their inability to provide low optical loss, compact size, high efficiency, and wide optical bandwidth, all at the same time. We report a thermo-optic silicon modulator realized in a 0.0023-mm2 silicon footprint of a commercial foundry silicon photonics process. The optical modulator consumes 2.56 mW for 180° phase modulation over 100-nm optical bandwidth while achieving 1.23-dB optical loss without air-gap trench or silicon undercut post-processing. Geometrical design optimization, at the core of this demonstration, is applicable to the realization of compact thermo-optic devices for large-scale programmable photonic integrated systems, with a potential to reduce power consumption roughly by an order of magnitude without sacrificing scalability and optical modulation bandwidth.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

2018 (6)

H. Jayatilleka, H. Shoman, R. Boeck, N. A. F. Jaeger, L. Chrostowski, and S. Shekhar, “Automatic configuration and wavelength locking of coupled silicon ring resonators,” J. Light. Technol. 36, 210–218 (2018).
[Crossref]

S. Chung, H. Abediasl, and H. Hashemi, “A monolithically integrated large-scale optical phased array in silicon-on-insulator CMOS,” IEEE J. Solid-State Circuits 53, 275–296 (2018).
[Crossref]

P. Dumais, D. J. Goodwill, D. Celo, J. Jiang, C. Zhang, F. Zhao, X. Tu, C. Zhang, S. Yan, J. He, M. Li, W. Liu, Y. Wei, D. Geng, H. Mehrvar, and E. Bernier, “Silicon photonic switch subsystem with 900 monolithically integrated calibration photodiodes and 64-fiber package,” J. Light. Technol. 36, 233–238 (2018).
[Crossref]

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alising, and D. Englund, “Linear programmable nanophotonic processors,” Optica 12, 1623–1631 (2018).
[Crossref]

M. Bahadori, A. Gazman, N. Janosik, S. Rumley, Z. Zhu, R. Polster, Q. Cheng, and K. Bergman, “Thermal rectification of integrated microheaters for microring resonators in silicon photonics platform,” J. Light. Technol. 36, 773–788 (2018).
[Crossref]

S.-H. Kim, J.-B. You, H.-W. Rhee, D. E. Yoo, D.-W. Lee, K. Yu, and H.-H. Park, “High-performance silicon MMI switch based on thermo-optic control of interference modes,” IEEE Photonics Technol. Lett. 30, 1427–1430 (2018).
[Crossref]

2017 (3)

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, “Coherent solid-state LIDAR with silicon photonic optical phased arrays,” Opt. Lett. 42, 4091–4094 (2017).
[Crossref] [PubMed]

2016 (1)

C. Sun, M. Wade, M. Georgas, S. Lin, L. Alloatti, B. Moss, R. Kumar, A. H. Atabaki, F. Pavanello, J. M. Shainline, J. S. Orcutt, R. J. Ram, M. Popović, and V. Stojanović, “A 45 nm CMOS-SOI monolithic photonics platform with bit-statistics-based resonant microring thermal tuning,” IEEE J. Solid-State Circuits 51, 893–907 (2016).
[Crossref]

2015 (4)

H. Abediasl and H. Hashemi, “Monolithic optical phased-array transceiver in a standard SOI CMOS process,” Opt. Express 23, 6509–6519 (2015).
[Crossref] [PubMed]

N. Dupuis, B. G. Lee, A. V. Rylyakov, D. M. Kuchta, C. W. Baks, J. S. Orcutt, D. M. Gill, W. M. J. Green, and C. L. Schow, “Design and fabrication of low-insertion-loss and low-crosstalk broadband 2×2 Mach-Zehnder silicon photonic switches,” J. Light. Technol. 33, 3597–3606 (2015).
[Crossref]

Z. Lu, K. Murray, H. Jayatilleka, and L. Chrostowski, “Michelson interferometer thermo-optic switch on SOI with a 50-μw power consumption,” IEEE Photonics Technol. Lett. 22, 2319–2322 (2015).

K. Murray, Z. Lu, H. Jayatilleka, and L. Chrostowski, “Dense dissimilar waveguide routing for highly efficient thermo-optic switches on silicon,” Opt. Express 23, 19575–19585 (2015).
[Crossref] [PubMed]

2014 (7)

S.-K. Ryu, J. Im, P. S. Ho, and R. Huang, “A kinetic decomposition process for air-gap interconnects and induced deformation instability of a low-k dielectric cap layer,” J. Mech. Sci. Technol. 28, 255–261 (2014).
[Crossref]

M. Nedeljkovic, S. Stanković, C. J. Mitchel, A. Z. Khokhar, D. J. T. Scott, A. Reynolds, F. Y. Gardes, C. G. Littlejohns, G. T. Reed, and G. Z. Mashanovich, “Mid-infrared thermo-optic modulators in SOI,” IEEE Photon. Technol. Lett. 26, 1352–1355 (2014).
[Crossref]

J. Sun, E. Timurdogan, A. Yaacobi, Z. Su, E. S. Hosseini, D. B. Cole, and M. R. Watts, “Large-scale silicon photonic circuits for optical phased arrays,” IEEE J. Sel. Top. on Quantum Electron. 20, 264–278 (2014).
[Crossref]

N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, “Efficient, compact and low loss thermo-optic phase shifter in silicon,” Opt. Express 22, 10487–10493 (2014).
[Crossref] [PubMed]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014).
[Crossref]

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 3541 (2014).
[Crossref] [PubMed]

A. Rickman, “The commercialization of silicon photonics,” Nat. Photonics 8, 579–582 (2014).
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2013 (3)

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Figures (10)

Fig. 1
Fig. 1 Performance trade-offs of thermo-optic silicon waveguide modulators at near-infrared without inherent optical bandwidth limitation. Data points from the published experimental results show that the geometrical design optimization demonstrated in this work improves the thermo-optic modulator performance trade-offs among optical loss, modulation bandwidth, and power consumption for 180° (π) phase shift. See Appendix for a feasible design space around the measured performance of this work, which can be explored with geometrical optimization.
Fig. 2
Fig. 2 Fabricated structure of a silicon thermo-optic modulator with geometrical design optimization for efficiency improvement. (a) Typical cross-section of the proposed silicon thermo-optic phase modulator with additional peripheral waveguides. (b) Typical cross-section of a conventional ultralow power thermo-optic phase modulator with air-gap trench and silicon undercut for increased heat isolation [31]. (c) Physical layout of the fabricated thermo-optic phase modulator on a silicon-on-insulator (SOI) wafer, which leverages geometrical design optimization for improving the energy efficiency of thermo-optic modulation for large-scale integration, with SEM photographs on multi-section Clothoid bend structures and waveguide array with alternating widths. (d) Microphotograph of a fabricated modulator test chip, which characterizes the modulator in a Mach-Zhender interferometer. (e) Vertical cross-section of the fabricated modulator on a SOI wafer (2 μm buried oxide (BOX) thickness) with simulated temperature profile. (f) Horizontal cross-section of the fabricated modulator with simulated temperature profile at the center of the silicon waveguides (the complete modulator structure is superimposed on the temperature profile as a guide).
Fig. 3
Fig. 3 Computational characterization. (a) Temperature profile in the middle of the fabricated thermo-optic modulator at the plane parallel to the waveguide center. (b) Power consumption necessary for 180° (π) phase modulation depending on modulator size, showing a relation between the thermo-optic modulation efficiency and the number of peripheral waveguides for blackgeometrical design optimization. Experimental result is plotted as an upward triangle point, simulated result is plotted as other discrete points, and analytical result is plotted as a dashed line (See Appendix B-1 for the first-order analytical mode to predict the modulator power consumption from waveguide temperature profile). (c) Transient step response of the fabricated thermo-optic modulator with 10-mW heater input, showing a relation between the modulation speed and the number of peripheral waveguides for geometrical design optimization. (d) Thermal crosstalk induced by an adjacent modulator depending on the number of peripheral waveguides for geometrical design optimization.
Fig. 4
Fig. 4 Experimental results. (a) Mach-Zhender interferometer (MZI) for the experimental characterization of the thermo-optic modulator. (b) Extinction ratio of −23.03 dB on average with three test chips for optical loss characterization, corresponding to the optical loss of 1.23 dB. (c) Power consumption of 2.56 mW for 180° thermo-optic phase modulation. (d) Normalized optical transmission, which is extracted from the measured MZI output power, for optical bandwidth characterization. (e) Thermo-optic modulation bandwidth of 10.1 kHz, which is measured in time domain by observing 10–90% rise and fall time. (f) Thermo-optic modulation bandwidth, which is measured in frequency domain by using a sinusoidal modulation signal.
Fig. 5
Fig. 5 Multi-Clothoid 180° bend waveguide structure in the fabricated thermo-optic modulator. 15 parallel straight waveguides with alternating widths in the modulator are connected by two types of 14 U-turn bend structures, each of which is constructed by four Clothoid bend waveguides (Segment 1–4), a tilted taper waveguide (Segment 5), and a straight strip waveguide (Segment 6). (a) bend waveguide structure, which is analytically drawn by using geometrical design equations derived in Appendix A-2, in the fabricated modulator. (b) Layout of a Clothoid bend waveguide segment 1. (c) Curvature change of the fabricated multi-Clothoid bend structure as a function of path length L starting from the segment 1. Note that the curvature of each Clothoid curve segment is zero at the point where the curve interfaces a straight waveguide or a taper waveguide. The curvature reaches the maximum at the point where the waveguide bend angle ϕ has the maximum value of θ.
Fig. 6
Fig. 6 Multi-Clothoid 180° bend waveguide design. (a) Analytically created bend structures with H = 4 μm, g = 1 μm, Wa = 500 nm, and Wb = 400 nm for a varying Clothoid bend angle θa. Four parameters (Ht, g, Wa, Wb) are predetermined by considering thermal and optical crosstalk while θa provides a degree of freedom to minimize the optical loss of the bend structure. (b) Experimentally measured optical loss of multi-Clothoid bend waveguides with θa = 105° (0.077 dB in average with 0.014 dB standard deviation from 5 samples) in comparison with simulation results. The inset SEM photographs show the two types of the bend structure (the measured optical loss is an average on the two types). (c) Comparison of the type-A multi-Clothoid 180° bend structure with conventional 180° bend structures based on standard circular bend waveguides. The improved type-A multi-Clothoid bend structure is described in Appendix A-4.
Fig. 7
Fig. 7 SEM photograph of a low-loss Clothoid bend waveguide that we designed such that the inner wall width (distance between the center line and the inner edge) continuously varies from 400 nm to 500 nm by following a Heaviside function described in [57].
Fig. 8
Fig. 8 Steady-state analytical model of the thermo-optic modulator with peripheral waveguides for geometrical design optimization. (a) Physical modulator structure and simplified cylindrical thermal analysis model, which represents an array of 2k +1 waveguides and a heater. Note that the heater overlaps with the center waveguide in the analysis model (this assumption simplifies the mathematical representation of the model). (b) Comparison on the thermo-optic modulator power consumption between analytical model prediction, 3D numerical device simulation, and experimental measurement result. Note that N represents the number of peripheral waveguides.
Fig. 9
Fig. 9 Transient analytical model of the thermo-optic modulator with geometrical design optimization. (a) Transient analytical model prediction on the waveguide temperature profile in comparison with numerical simulation. (b) Transient analytical model prediction on the optical phase response in comparison with numerical simulation and experimental measurement.
Fig. 10
Fig. 10 Experimental setup for characterizing steady-state and transient thermo-optic phase response.

Tables (5)

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Table 1 Comparison of Experimental Results with Previously Reported Near-Infrared Silicon Thermo-Optic Modulators.

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Table 2 Geometrical Parameters of a Multi-Clothoid 180° Bend Waveguide.

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Table 3 Geometrical Design Equations to Create a Multi-Clothoid 180° Bend Waveguide (see Eq. (20) for Obx) and Eqs. (27)(30) for Ab).

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Algorithm 1 Multi-Clothoid 180° bend waveguide design procedure under a physical size constraint.

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Table 4 Geometrical Design Parameters of the Multi-Clothoid 180° Bend Waveguide Structures in the Fabricated Thermo-Optic Modulator.

Equations (64)

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d l = R ( ϕ ) d ϕ , d x = d l cos ϕ , d y = d l sin ϕ .
R ( ϕ ) L ( ϕ ) = A 2
L ( ϕ ) = 0 ϕ R ( ψ ) d ψ .
d l = A 2 L ( ϕ ) d ϕ 0 L ( ϕ ) l d l = 0 ϕ A 2 d ψ L ( ϕ ) 2 2 = A 2 ϕ ,
L ( ϕ ) = A 2 ϕ
d x = ( A 2 ϕ cos ϕ ) d ϕ , d y = ( A 2 ϕ sin ϕ ) d ϕ .
x ( ϕ ) = A 2 0 ϕ cos ψ ψ d ψ = A 2 0 ϕ 1 ψ k = 0 ( 1 ) k ( 2 k ) ! ψ 2 k d ψ = A 2 k = 0 [ ( 1 ) k ( 2 k ) ! 0 ϕ ψ 2 k 1 2 d ψ ] = A 2 k = 0 [ ( 1 ) k ( 2 k + 1 2 ) ( 2 k ) ! ϕ 2 k + 1 2 ]
y ( ϕ ) = A 2 0 ϕ sin ψ ψ d ψ = A 2 0 ϕ 1 ψ k = 0 ( 1 ) k ( 2 k + 1 ) ! ψ 2 k + 1 d ψ = A 2 k = 0 [ ( 1 ) k ( 2 k + 1 ) ! 0 ϕ ψ 2 k + 1 2 d ψ ] = A 2 k = 0 [ ( 1 ) k ( 2 k + 3 2 ) ( 2 k + 1 ) ! ϕ 2 k + 3 2 ] .
P i = [ x ( ϕ ) W 2 sin ϕ y ( ϕ ) + W 2 cos ϕ ] , P o = [ x ( ϕ ) + W 2 sin ϕ y ( ϕ ) W 2 cos ϕ . ]
P 1 S ( ϕ ) = M [ x 1 ( ϕ ) y 1 ( ϕ ) ] = [ x 1 ( ϕ ) y 1 ( ϕ ) ] = [ y 1 ( ϕ ) x 1 ( ϕ ) ]
P 1 A ( ϕ ) = P 1 S ( ϕ ) + P 1 ( θ a ) P 1 T ( θ a ) = [ y 1 ( ϕ ) + x 1 ( θ a ) y 1 ( θ a ) x 1 ( ϕ ) + y 1 ( θ a ) x 1 ( θ a ) ] .
P 2 ( ϕ ) = [ x 2 ( ϕ ) y 2 ( ϕ ) ] T = J ( P 1 A ( ϕ E ) + E = [ x 1 ( ϕ ) sin δ a + y 1 ( ϕ ) cos δ a + x 1 ( θ a ) [ 1 + sin δ a ] y 1 ( θ a ) cos δ a x 1 ( ϕ ) cos δ a + y 1 ( ϕ ) sin δ a x 1 ( θ a ) cos δ a + y 1 ( θ a ) [ 1 sin δ a ] ] .
J = [ cos δ a sin δ a sin δ a cos δ a ] .
δ a = 2 θ a 3 π 2 .
P 3 ( ϕ ) = [ A b 2 k = 0 ( ( 1 ) k ( 2 k + 1 2 ) ( 2 k ) ! ϕ 2 k + 1 2 ) A b 2 k = 0 ( ( 1 ) k ( 2 k + 3 2 ) ( 2 k + 1 ) ! ϕ 2 k + 3 2 ) ] , P 4 ( ϕ ) = [ x 3 ( ϕ ) sin δ b + y 3 ( ϕ ) cos δ b + x 3 ( θ b ) [ 1 + sin δ b ] y 3 ( θ b ) cos δ b x 3 ( ϕ ) cos δ b + y 3 ( ϕ ) sin δ b x 3 ( θ b ) cos δ b + y 3 ( θ b ) [ 1 sin δ b ] ] .
θ b = θ a π 2 .
O b = [ O b x O b y ] = [ x 4 ( 0 ) + x 2 ( 0 ) L t cos ( 2 θ b ) g ]
P 3 ( ϕ ) = P 3 ( ϕ ) + O b = [ x 3 ( ϕ ) + O b x y 3 ( ϕ ) + g ] ,
P 4 ( ϕ ) = P 4 ( ϕ ) + O b = [ x 4 ( ϕ ) + O b x y 4 ( ϕ ) + g ] ,
O b x = [ x 3 ( θ b ) ( 1 + sin δ b ) y 3 ( θ b ) cos δ b ] + [ x 1 ( θ a ) ( 1 + sin δ a ) y 1 ( θ a ) cos δ a ] L t cos ( 2 θ b ) = x 3 ( θ a π / 2 ) ( 1 cos ( 2 θ a ) ) + y 3 ( θ a π / 2 ) sin ( 2 θ a ) + x 1 ( θ a ) ( 1 + cos ( 2 θ a ) ) + y 1 ( θ a ) sin ( 2 θ a ) + L t cos ( 2 θ a ) .
P 5 ( l ) = [ x 5 ( l ) y 5 ( l ) ] = [ x 4 ( 0 ) + l sin ( 2 θ b ) y 4 ( 0 ) + l cos ( 2 θ b ) ] .
P 5 i ( l ) = P 5 ( l ) W e ( l ) , P 5 o ( l ) = P 5 ( l ) + W e ( l )
W e ( l ) = [ ( W b + ( W a W b ) / L t 2 ) cos ϕ ( W b + ( W a W b ) / L t 2 ) sin ϕ ]
d y 2 ( ϕ ) d x 2 ( ϕ ) = d y 2 ( ϕ ) d ϕ / d x 2 ( ϕ ) d ϕ = 0 ,
d y 2 ( ϕ ) d ϕ = cos δ a d x 1 ( ϕ ) d ϕ + sin δ a d y 1 ( ϕ ) d ϕ = cos δ a ( A 2 ϕ cos ϕ ) + sin δ a ( A 2 ϕ sin ϕ ) = A 2 ϕ cos ( δ a ϕ ) = 0 .
ϕ = δ a + π 2 = 2 θ a π ,
H = y 2 ( ϕ ) = [ y 2 ( ϕ ) y 2 ( θ a ) ] + y 1 ( θ a ) = [ x 1 ( 2 θ a π ) x 1 ( θ a ) ] cos δ a + [ y 1 ( 2 θ a π ) y 1 ( θ a ) ] sin δ a + y 1 ( θ a ) = A a 2 k = 0 ( 1 ) k ( ( 2 θ a π ) 2 k + 1 2 θ a 2 k + 1 2 ) ( 2 k + 1 2 ) ( 2 k + 1 ) ! cos δ a + A a 2 k = 0 ( 1 ) k [ ( ( 2 θ a π ) 2 k + 3 2 θ a 2 k + 3 2 ) sin δ a + θ a 2 k + 3 2 ] ( 2 k + 3 2 ) ( 2 k + 1 ) !
h 2 = H g L t sin ( 2 θ b ) [ y 2 ( ϕ ) y 2 ( 0 ) ] = [ x 1 ( θ a ) cos δ a + y 1 ( θ a ) ( 1 sin δ a ) ] g + L t sin ( 2 θ a ) ,
h 2 = g + L t sin ( 2 θ a ) + A a 2 k = 0 [ ( 1 ) 2 sin ( 2 θ a ) ( 2 k + 1 2 ) ( 2 k ) ! θ a 2 k + 1 2 ] + A a 2 k = 0 [ ( 1 ) k ( 1 cos 2 θ a ) ( 2 k + 3 2 ) ( 2 k + 1 ) ! θ a 2 k + 3 2 ]
h 2 = y 4 ( 0 ) = x 3 ( θ b ) cos ( δ b ) + y 3 ( θ b ) ( 1 sin δ b ) = A b 2 k = 0 [ ( 1 ) k sin ( 2 θ a ) ( 2 k + 1 2 ) ( 2 k ) ! ( θ a π 2 ) 2 k + 1 2 ] + A b 2 k = 0 [ ( 1 ) k ( 1 + cos 2 θ a ) ( 2 k + 3 2 ) ( 2 k + 1 ) ! ( θ a π 2 ) 2 k + 3 2 ] .
L u = 2 ( A a 2 θ a + A b 2 θ b + L t + O b x )
( k T ) + q ˙ = ρ c p T t ,
1 r r ( k r T ( r ) r ) = 0
T ( r i ) = T i , T ( r o ) = T o .
T ( r ) = T i T i T o ln ( r o ) ln ( r i ) ( ln ( r ) ln ( r i ) ) = c 1 ln ( r ) + c 2
c 1 = T i T o ln ( r o ) ln ( r i ) , c 2 = T i ln ( r o ) T o ln ( r i ) ln ( r o / r i ) .
q h = k A d T d r = k ( 2 π r L ) d T d r ,
q h = 2 π k L ( T i T o ) ln ( r o ) ln ( r i ) ,
T i = T o + ln ( r o / r i ) 2 π k L q h .
ϕ c = ( 2 π L λ ) d n d T Δ T = ( 2 π L λ ) d n d t ( T i T o ) .
ϕ c = d n d T ( ln ( r o / r i ) k λ ) q h .
q h , r = ( k λ ln ( r o / r i ) ) ϕ c d n d T = ( k λ ln ( r o / r i ) ) π d n d T .
ϕ m = m = k k ( 2 π L λ ) d n d T ( T | m | p ) T o ) = ϕ c + 2 m = 1 k ( 2 π L λ ) d n d T ( T ( mp ) T o ) = ϕ c + 2 m = 1 k ( 2 π L λ ) d n d T ( ln ( r o ) ln ( mp ) ln ( r o ) ln ( r i ) ( T i T o ) ) .
ϕ e = ϕ m ϕ c = ( 4 π L λ ) d n d T T i T o ln ( r o / r i ) ln ( ( r o / p ) k k ! ) .
q h , π = ( k λ ln ( r o / r i ) ) ϕ c d n d T = ( k λ ln ( r o / r i ) ) π ϕ e d n d T .
Δ q h , r = q h , r q h , r = ( 4 π k L ) T i T o ln 2 ( r o / r i ) ln ( ( r o / p ) k k ! ) .
1 r r ( r T ( r , t ) r ) = 1 α T ( r , t ) t
T ( r , 0 ) = T i T o ln ( r o / r i ) ln ( r ) + T i ln ( r o ) T o ln ( r i ) ln ( r o / r i ) .
1 r Γ ( r ) d d r ( r d Γ ( r ) d r ) = 1 α Λ ( t ) d Λ ( t ) d t .
1 r d d r ( r d Γ ( r ) d r ) = ψ 2
1 α d Λ ( t ) d t = ψ 2
x 2 d 2 Γ d x 2 + x d Γ d r + x 2 Γ = 0 ,
Γ ( r ) = J 0 ( ψ r ) = m = 0 ( 1 ) m ( m ! ) 2 ( ψ r 2 ) 2 m .
d Λ d t + α ψ 2 Λ = 0
Λ ( t ) = C e α ψ 2 t
T ( r , t ) = n = 1 Γ n ( r ) Λ n ( r ) = n = 1 J 0 ( ψ n r ) C n e α ψ n 2 t .
T ( r , 0 ) = n = 1 C n J 0 ( ψ n r ) .
C n = 2 H ( ψ n ) r o 2 J 1 2 ( ψ n r o )
J 1 ( ψ n r o ) = m = 0 ( 1 ) m m ! ( m + 1 ) ! ( ψ n r o 2 ) 2 m + 1 ,
H ( ψ n ) = 0 r J 0 ( ψ n r ) T ( r , 0 ) d r .
T ( r , t ) = 2 r o 2 n = 1 [ J 0 ( ψ n r ) J 1 2 ( ψ n r o ) e α ψ n 2 t 0 r o r J 0 ( ψ n r ) T ( r , 0 ) d r ] .
T h ( r , t ) = T ( r , 0 ) T ( r , t ) = n = 1 C n J 0 ( ψ n r ) ( 1 e α ψ n 2 t ) .
ϕ m ( t ) = m = k k ( 2 π L λ ) d n d T T h ( | m | p , t ) = ( 4 π L λ r o 2 ) d n d T m = k k n = 1 [ J 0 ( ψ n | m | p ) J 1 2 ( ψ n r o ) ( 1 e α ψ n 2 t ) 0 r o | m | p J 0 ( ψ n | m | p ) T ( r , 0 ) d r ]
q h ( t ) = 2 π k c L ( T s ( 0 , ) ln ( r o / r i ) u ( t ) = 2 π k c L ln ( r o / r i ) [ T o + n = 1 C n ] u ( t )

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