Abstract

We propose a novel waveguide self-coupling based reconfigurable resonance structure to work as a flat-top second-order tunable filter and a tunable delay line with low group delay dispersion. The high-order resonance features result from the mutual mode coupling between the clockwise and counter-clockwise resonance eigenmodes. The transfer-matrix method is used to theoretically analyze the device optical performances. The relations between the two embedded phase shifters for achieving flat-top filtering and group delay responses are given. As the coupled resonances are provided by only one physical resonator, the device is inherently more compact and resilient to fabrication errors compared to conventional microring resonators. Phase tuning for its reconfiguration is also simpler and more power-efficient.

© 2011 OSA

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    [CrossRef]

2011 (1)

2010 (9)

X. Zheng, I. Shubin, G. Li, T. Pinguet, A. Mekis, J. Yao, H. Thacker, Y. Luo, J. Costa, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “A tunable 1x4 silicon CMOS photonic wavelength multiplexer/demultiplexer for dense optical interconnects,” Opt. Express 18(5), 5151–5160 (2010).
[CrossRef] [PubMed]

J. E. Cunningham, I. Shubin, X. Zheng, T. Pinguet, A. Mekis, Y. Luo, H. Thacker, G. Li, J. Yao, K. Raj, and A. V. Krishnamoorthy, “Highly-efficient thermally-tuned resonant optical filters,” Opt. Express 18(18), 19055–19063 (2010).
[CrossRef] [PubMed]

X. Luo, H. Chen, and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonators,” Opt. Lett. 35(17), 2940–2942 (2010).
[CrossRef] [PubMed]

P. Dong, N. N. Feng, D. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express 18(23), 23784–23789 (2010).
[CrossRef] [PubMed]

J. Cardenas, M. A. Foster, N. Sherwood-Droz, C. B. Poitras, H. L. Lira, B. Zhang, A. L. Gaeta, J. B. Khurgin, P. Morton, and M. Lipson, “Wide-bandwidth continuously tunable optical delay line using silicon microring resonators,” Opt. Express 18(25), 26525–26534 (2010).
[CrossRef] [PubMed]

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

2009 (6)

2008 (2)

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

2007 (3)

2006 (2)

2005 (3)

2002 (1)

J. Heebner and R. Boyd, “'Slow' and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

2001 (1)

G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[CrossRef]

1998 (1)

1997 (2)

B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. 22(1), 4–6 (1997).
[CrossRef] [PubMed]

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

1995 (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

1987 (1)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Adibi, A.

Agarwal, A.

Asghari, M.

Banwell, T.

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Bergman, K.

Biberman, A.

Boyd, R.

J. Heebner and R. Boyd, “'Slow' and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

Canciamilla, A.

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

Cardenas, J.

Chen, H.

Chen, W.

Chin, M.

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

Chu, S.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Chu, S. T.

Costa, J.

Cunningham, J. E.

Dahlem, M. S.

Darmawan, S.

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

De La Rue, R.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

Dong, P.

Eggleton, B.

G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[CrossRef]

Etemad, S.

Feng, D.

Feng, N. N.

Ferrari, C.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

Foresi, J.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Foster, M. A.

Franck, T.

Gaeta, A. L.

Gardes, F.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Gondarenko, A.

Green, W.

Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

Haus, H.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Haus, H. A.

Heebner, J.

J. Heebner and R. Boyd, “'Slow' and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

Hodge, D.

Holzwarth, C. W.

Ippen, E. P.

Jackel, J.

Kärtner, F. X.

Keil, U.

Khilo, A.

Khurgin, J. B.

Krishnamoorthy, A. V.

Laine, J.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Laine, J. P.

Lee, B.

Lee, D. C.

Lenz, G.

G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[CrossRef]

Li, G.

Li, Q.

Liang, H.

Liao, L.

Lim, D.

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

Lipson, M.

Lira, H. L.

Little, B.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Little, B. E.

Liu, A.

Luff, B. J.

Luo, X.

Luo, Y.

Madsen, C.

G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[CrossRef]

Manipatruni, S.

Manolatou, C.

Martinelli, M.

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

Mashanovich, G.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Mei, T.

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

Mekis, A.

Melloni, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

Menendez, R.

Morichetti, F.

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

Morse, M.

Morton, P.

Morton, P. A.

Okamoto, K.

L. Zhou, K. Okamoto, and S. Yoo, “Athermalizing and trimming of slotted silicon microring resonators with UV-sensitive PMMA upper-cladding,” IEEE Photon. Technol. Lett. 21(17), 1175–1177 (2009).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Pinguet, T.

Poitras, C.

Poitras, C. B.

Poon, A. W.

Popovic, M.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Preston, K.

Qian, W.

Qiu, M.

Raj, K.

Reed, G.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Rubin, D.

Samara-Rubio, D.

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[CrossRef]

Sherwood-Droz, N.

Shubin, I.

Slusher, R.

G. Lenz, B. Eggleton, C. Madsen, and R. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[CrossRef]

Smith, H. I.

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Soltani, M.

Soref, R.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Sorel, M.

F. Morichetti, A. Canciamilla, M. Martinelli, A. Samarelli, R. De La Rue, M. Sorel, and A. Melloni, “Coherent backscattering in optical microring resonators,” Appl. Phys. Lett. 96(8), 081112 (2010).
[CrossRef]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[CrossRef]

Su, Y.

Thacker, H.

Thomson, D.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Tobing, L.

L. Tobing, S. Darmawan, D. Lim, M. Chin, and T. Mei, “Relaxation of Critical Coupling Condition and Characterization of Coupling-Induced Frequency Shift in Two-Ring Structures,” IEEE J. Sel. Top. Quantum Electron. 16(1), 77–84 (2010).
[CrossRef]

Toliver, P.

Torregiani, M.

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[CrossRef]

Nat. Photonics (3)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[CrossRef]

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[CrossRef]

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Nature (1)

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Other (1)

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Figures (6)

Fig. 1
Fig. 1

Schematic of the waveguide self-coupling based reconfigurable resonance structure. Electric fields before and after each coupler are labeled. The large blue and small red arrows indicate clockwise (CW) and counter-clockwise (CCW) resonance loops, respectively.

Fig. 2
Fig. 2

(a) and (b) Transmission intensity and group delay responses for the device configured as a notch filter. (c) and (d) Transmission and reflection intensity and group delay responses for the device configured as an add-drop filter.

Fig. 3
Fig. 3

(a) Relation between Δϕx and Δϕio for flap-top filters. (b) Solid curves: reflection intensity spectra of our device; dotted curves: drop-port intensity spectra of an add-drop filter. (c) FSR-normalized transition and reflection bandwidths as functions of Δϕio . (d) Insertion loss and out-of-band rejection ratio as functions of Δϕio . In (b)-(d), the flap-top passband condition is satisfied.

Fig. 4
Fig. 4

(a) Relation between Δϕx and Δϕio for flap-top group delay responses. (b)-(d) Insertion loss, normalized group delay bandwidth, and group delay change as a function of Δϕio . (e) and (f) Typical reflection intensity and group delay spectra. In (b)-(f), the flat-top group delay condition is satisfied.

Fig. 5
Fig. 5

Comparison between our device and a double ring ADF delay line in terms of total phase shift versus group delay.

Fig. 6
Fig. 6

(a)-(c) Reflection intensity spectra for various input and output coupling ratios. (d)-(f) Reflection group delay responses for various input and output coupling ratios.

Equations (24)

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[ E a 2 + E a 4 + E b 2 + E b 4 + ] = M c i o [ E a 1 + E a 3 + E b 1 + E b 3 + ] = [ t a i κ a 0 0 i κ a t a 0 0 0 0 t b i κ b 0 0 i κ b t b ] [ E a 1 + E a 3 + E b 1 + E b 3 + ] ,
[ E c 2 E c 1 + E c 3 + E c 4 ] = M t 1 [ E a 2 + E a 4 + E b 2 + E b 4 + ] = [ e i ( n e f f k 0 L 3 + Δ ϕ i o ) 0 0 0 0 e i n e f f k 0 L 2 0 0 0 0 e i n e f f k 0 L 2 0 0 0 0 e i ( n e f f k 0 L 3 Δ ϕ i o ) ] [ E a 2 + E a 4 + E b 2 + E b 4 + ] ,
n e f f = n e f f ' i n e f f ' '
n e f f '
n e f f ' '
[ E c 2 + E c 1 E c 3 E c 4 + ] = M c x [ E c 2 E c 1 + E c 3 + E c 4 ] = [ 0 i sin ( Δ ϕ x ) i cos ( Δ ϕ x ) 0 i sin ( Δ ϕ x ) 0 0 i cos ( Δ ϕ x ) i cos ( Δ ϕ x ) 0 0 i sin ( Δ ϕ x ) 0 i cos ( Δ ϕ x ) i sin ( Δ ϕ x ) 0 ] [ E c 2 E c 1 + E c 3 + E c 4 ] ,
[ E a 1 + E a 3 + E b 1 + E b 3 + ] = M t 2 [ E a 1 + E a 3 E b 1 E b 3 + ] = [ 1 0 0 0 0 0 e i n e f f k 0 L 1 0 0 e i n e f f k 0 L 1 0 0 0 0 0 1 ] [ E a 1 + E a 3 E b 1 E b 3 + ] .
[ E a 1 E a 3 E b 1 E b 3 ] = M [ E a 1 + E a 3 E b 1 E b 3 + ] = M c i o M t 1 M c x M t 1 M c i o M t 2 [ E a 1 + E a 3 E b 1 E b 3 + ] .
H t ( z ) = E b 3 E i n = t w g t x t i o t x ( t i o 2 + 1 ) a z 1 + t i o a 2 z 2 1 2 t x t i o a z 1 + ( κ x 2 + t x 2 t i o 2 ) a 2 z 2 ,
H r ( z ) = E a 1 E i n = e i Δ ϕ i o t w g κ x 1 2 t x t i o a z 1 + a 2 z 2 1 2 t x t i o a z 1 + ( κ x 2 + t x 2 t i o 2 ) a 2 z 2 ,
H t ( z ) = t w g t i o a z 1 1 t i o a z 1 ,
H t ( z ) = t w g t x 2 a z 1 1 + κ x 2 a 2 z 2 ,
H r ( z ) = e i Δ ϕ i o t w g κ x + κ x a 2 z 2 1 + κ x 2 a 2 z 2 ,
ϕ p 0 = tan 1 ( κ x t x t i o ) ,
γ = ln ( t x 2 t i o 2 + κ x 2 ) 2 ln a .
κ i o > 2 ( κ x / t x + ln a ) / t x .
p 0 = | p 0 | e i ϕ p 0 = a ( t x t i o + i κ x ) ,
z 0 = | z 0 | e i ϕ z 0 = a [ t x t i o + i 1 ( t x t i o ) 2 ] .
( z 0 * + z 0 ) | 1 z 0 | 2 | z 0 z 0 * | 2 | 1 z 0 | 4 = ( p 0 * + p 0 ) | 1 p 0 | 2 | p 0 p 0 * | 2 | 1 p 0 | 4 .
τ g | ϕ = 0 = 2 T R [ 1 r 0 1 + | p 0 | 2 2 r 0 1 r 0 1 + | z 0 | 2 2 r 0 ] ,
| 1 z 0 1 p 0 | 6 = 2 r 0 2 + ( | z 0 | 2 + 1 ) r 0 4 | z 0 | 2 2 r 0 2 + ( | p 0 | 2 + 1 ) r 0 4 | p 0 | 2 1 | z 0 | 2 1 | p 0 | 2 .
2 r 0 2 + ( | p 0 | 2 + 1 ) r 0 4 | p 0 | 2 = 0.
1 t x t i o κ x = 4 t x t i o 1 3 .
Δ ϕ x 3 3 Δ ϕ i o 2 .

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