Abstract

The time-reversed second-harmonic generation in one-dimensional nonlinear photonic crystals has been theoretically studied without the undepleted pump approximation. A simple criterion has been deduced which determines the energy flow. Based on it, two kinds of structures with different symmetries are presented to realize the nonlinear time reversal effect. A completely reciprocal nonlinear response is also found in the same process. Furthermore, a multi-section-cascaded structure is proposed to realize the nonlinear time reversal at any given position.

© 2017 Optical Society of America

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References

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  2. M. Fink, “Time-reversal mirrors,” J. Phys. D Appl. Phys. 26, 1333–1350 (1993).
  3. M. I. Stockman and X. Li, “Highly efficient spatio-temporal coherent control in nanoplasmonics on nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008).
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    [PubMed]
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  8. J. B. Pendry, “Time reversal and negative refraction,” Science 322(5898), 71–73 (2008).
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2016 (2)

M. Yachini, B. Malomed, and A. Bahabad, “Envelope time reversal of optical pulses following frequency conversion with accelerating quasi-phase-matching,” ACS Photonics 3, 2017–2021 (2016).

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

2013 (1)

Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixing in nonlinear optics,” Sci. Rep. 3, 3245 (2013).
[PubMed]

2011 (2)

S. Longhi, “Time-reversed optical parametric oscillation,” Phys. Rev. Lett. 107(3), 033901 (2011).
[PubMed]

Y. Sivan and J. B. Pendry, “Time reversal in dynamically tuned zero-gap periodic systems,” Phys. Rev. Lett. 106(19), 193902 (2011).
[PubMed]

2010 (1)

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

2009 (1)

2008 (2)

M. I. Stockman and X. Li, “Highly efficient spatio-temporal coherent control in nanoplasmonics on nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008).

J. B. Pendry, “Time reversal and negative refraction,” Science 322(5898), 71–73 (2008).
[PubMed]

2007 (2)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[PubMed]

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026606 (2007).
[PubMed]

2004 (4)

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93(17), 173903 (2004).
[PubMed]

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

2003 (1)

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).

2002 (1)

2001 (1)

M. Fink and C. Prada, “Acoustic time-reversal mirrors,” Inverse Probl. 17, R1–R38 (2001).

2000 (2)

R. Carminati, J. J. Sáenz, J. J. Greffet, and M. Nietovesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).

D. Marom, D. Panasenko, R. Rokitski, P. C. Sun, and Y. Fainman, “Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25(2), 132–134 (2000).
[PubMed]

1993 (2)

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photonics Technol. Lett. 5, 92–95 (1993).

M. Fink, “Time-reversal mirrors,” J. Phys. D Appl. Phys. 26, 1333–1350 (1993).

1982 (1)

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).

1980 (1)

1979 (1)

1978 (1)

D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett. 33, 805–807 (1978).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).

Bahabad, A.

M. Yachini, B. Malomed, and A. Bahabad, “Envelope time reversal of optical pulses following frequency conversion with accelerating quasi-phase-matching,” ACS Photonics 3, 2017–2021 (2016).

Bennett, R. J.

D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett. 33, 805–807 (1978).

Berryman, J. G.

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[PubMed]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).

Borderies, P.

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).

Carminati, R.

R. Carminati, J. J. Sáenz, J. J. Greffet, and M. Nietovesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).

Chambers, D. H.

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[PubMed]

Chen, X.

Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixing in nonlinear optics,” Sci. Rep. 3, 3245 (2013).
[PubMed]

Chikama, T.

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photonics Technol. Lett. 5, 92–95 (1993).

Chumak, A. V.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

de Rosny, J.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

Derode, A.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).

Fainman, Y.

Fan, S.

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93(17), 173903 (2004).
[PubMed]

Fekete, D.

Fink, M.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

M. Fink and C. Prada, “Acoustic time-reversal mirrors,” Inverse Probl. 17, R1–R38 (2001).

M. Fink, “Time-reversal mirrors,” J. Phys. D Appl. Phys. 26, 1333–1350 (1993).

Foster, M. A.

Gaeta, A. L.

Greffet, J. J.

R. Carminati, J. J. Sáenz, J. J. Greffet, and M. Nietovesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).

Gregg, J. F.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Grischkowsky, D.

D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett. 33, 805–807 (1978).

Hillebrands, B.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Hong, X. H.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Karenowska, A. D.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Kuzucu, O.

Lerosey, G.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

Li, X.

M. I. Stockman and X. Li, “Highly efficient spatio-temporal coherent control in nanoplasmonics on nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008).

Lipson, M.

Liu, X.

Longhi, S.

S. Longhi, “Time-reversed optical parametric oscillation,” Phys. Rev. Lett. 107(3), 033901 (2011).
[PubMed]

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026606 (2007).
[PubMed]

Lu, R. E.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Malomed, B.

M. Yachini, B. Malomed, and A. Bahabad, “Envelope time reversal of optical pulses following frequency conversion with accelerating quasi-phase-matching,” ACS Photonics 3, 2017–2021 (2016).

Marom, D.

Meenakshi, S.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).

Mehendale, S. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).

Micolau, G.

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).

Miller, D. A.

Montaldo, G.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

Naito, T.

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photonics Technol. Lett. 5, 92–95 (1993).

Nietovesperinas, M.

R. Carminati, J. J. Sáenz, J. J. Greffet, and M. Nietovesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).

Okawachi, Y.

Panasenko, D.

Pendry, J. B.

Y. Sivan and J. B. Pendry, “Time reversal in dynamically tuned zero-gap periodic systems,” Phys. Rev. Lett. 106(19), 193902 (2011).
[PubMed]

J. B. Pendry, “Time reversal and negative refraction,” Science 322(5898), 71–73 (2008).
[PubMed]

Pepper, D. M.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).

Prada, C.

M. Fink and C. Prada, “Acoustic time-reversal mirrors,” Inverse Probl. 17, R1–R38 (2001).

Qin, Y. Q.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Ren, H.

Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixing in nonlinear optics,” Sci. Rep. 3, 3245 (2013).
[PubMed]

Rokitski, R.

Rustagi, K. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).

Sáenz, J. J.

R. Carminati, J. J. Sáenz, J. J. Greffet, and M. Nietovesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).

Saillard, M.

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).

Salem, R.

Serga, A. A.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Shiren, N. S.

D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett. 33, 805–807 (1978).

Sivan, Y.

Y. Sivan and J. B. Pendry, “Time reversal in dynamically tuned zero-gap periodic systems,” Phys. Rev. Lett. 106(19), 193902 (2011).
[PubMed]

Slavin, A. N.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Stockman, M. I.

M. I. Stockman and X. Li, “Highly efficient spatio-temporal coherent control in nanoplasmonics on nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008).

Sun, P. C.

Tiberkevich, V. S.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1, 141 (2010).
[PubMed]

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[PubMed]

Turner-Foster, A. C.

Wan, W.

Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixing in nonlinear optics,” Sci. Rep. 3, 3245 (2013).
[PubMed]

Watanabe, S.

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photonics Technol. Lett. 5, 92–95 (1993).

Yachini, M.

M. Yachini, B. Malomed, and A. Bahabad, “Envelope time reversal of optical pulses following frequency conversion with accelerating quasi-phase-matching,” ACS Photonics 3, 2017–2021 (2016).

Yang, B.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Yanik, M. F.

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93(17), 173903 (2004).
[PubMed]

Yariv, A.

Yue, Y. Y.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Zhang, C.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

Zhang, H.

Zhang, M.

Zheng, Y.

Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixing in nonlinear optics,” Sci. Rep. 3, 3245 (2013).
[PubMed]

Zhu, Y. Y.

B. Yang, Y. Y. Yue, R. E. Lu, X. H. Hong, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Rigorous intensity and phase-shift manipulation in optical frequency conversion,” Sci. Rep. 6, 27457 (2016).
[PubMed]

ACS Photonics (1)

M. Yachini, B. Malomed, and A. Bahabad, “Envelope time reversal of optical pulses following frequency conversion with accelerating quasi-phase-matching,” ACS Photonics 3, 2017–2021 (2016).

Appl. Phys. Lett. (1)

D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett. 33, 805–807 (1978).

IEEE J. Quantum Electron. (1)

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).

IEEE Photonics Technol. Lett. (1)

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Figures (4)

Fig. 1
Fig. 1 A schematic of the time-reversed and reciprocal SHG process in a 1D nonlinear photonic crystal with a defect embedded.
Fig. 2
Fig. 2 Simulations of the propagating processes of FW and SHW under the different phase mismatching conditions, where phase mismatches are (a) in even symmetry and (b) in odd symmetry.
Fig. 3
Fig. 3 The SHW output distribution with ΔkL/2 and Δ k ' L/2 being controlled in the range of ( π,π ).The phase shift induced by the defect is (a) Δφ=2[ ΔkL+2 ϕ 1 ( L ) ϕ 2 ( L ) ], (b) Δφ=π2ΔkL.
Fig. 4
Fig. 4 The generation of time-reversed replicas at any given position in the multi-section-cascaded defective NPC structure. (a) The phase mismatches are the same in different structure sections; (b) The sign of the phase mismatches is flipped every other section.

Equations (11)

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{ d A 1 dx =i K 1 f( x ) A 2 A 1 * exp( iΔ k 1 x ) d A 2 dx = 1 2 i K 1 f( x ) A 1 2 exp( iΔ k 1 x )
{ d A 1 dx =iK A 2 A 1 * exp( iΔkx ) d A 2 dx = 1 2 i K A 1 2 exp( iΔkx )
{ d A 1 dx =i K ' A 2 A 1 * exp( iΔ k ' x ) d A 2 dx = 1 2 i K ' A 1 2 exp( iΔ k ' x )
{ A 1 = y 1 exp( i ϕ 1 ) A 2 = y 2 exp( i ϕ 2 )
d I 2 dx | x=Ld = d I 2 dx | x=L+d
d I 2 dx = A 2 d A 2 dx + A 2 d A 2 dx
{ d A 2 dx | x=Ld = 1 2 i K A 1 2 exp[ iΔk( Ld ) ] d A 2 dx | x=Ld = 1 2 iK A 1 2 exp[ iΔk( Ld ) ]
{ d A 2 dx | x=L+d = 1 2 i K ' A 1 2 exp[ iΔ k ' ( L+d ) ] d A 2 dx | x=L+d = 1 2 i K ' A 1 2 exp[ iΔ k ' ( L+d ) ]
e i( 2 ϕ 1 ϕ 2 ) e iΔk( Ld ) e i( ϕ 2 2 ϕ 1 ) e iΔk( Ld ) = e iΔφ e i( ϕ 2 2 ϕ 1 ) e iΔ k ' ( L+d ) e iΔφ e i( 2 ϕ 1 ϕ 2 ) e iΔ k ' ( L+d )
Δφ=2[ ΔkL+2 ϕ 1 ( L ) ϕ 2 ( L ) ]
Δφ=π2ΔkL

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