Abstract

We analyze the sensitivity to inertial rotations Ω of a micron scale integrated gyroscope consisting of a coupled resonator optical waveguide (CROW). We show here that by periodic modulation of the evanescent coupling between resonators, the sensitivity to rotations can be enhanced by a factor up to 109 in comparison to a conventional CROW with uniform coupling between resonators. Moreover, the overall shape of the transmission through this CROW superlattice is qualitatively changed resulting in a single sharp transmission resonance located at Ω = 0s−1 instead of a broad transmission band. The modulated coupling therefore allows the CROW gyroscope to operate without phase biasing and with sensitivities suitable for inertial navigation even with the inclusion of resonator losses.

© 2011 Optical Society of America

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References

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  1. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
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  3. F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
    [Crossref]
  4. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
    [Crossref]
  5. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24, 1843–1849 (2006).
    [Crossref]
  6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006).
    [Crossref] [PubMed]
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    [Crossref]
  13. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
    [Crossref] [PubMed]
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  20. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004).
    [Crossref] [PubMed]
  21. S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.
  22. P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).
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2011 (3)

2010 (3)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
[Crossref]

2009 (2)

M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
[Crossref]

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

2007 (4)

2006 (6)

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006).
[Crossref] [PubMed]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24, 1843–1849 (2006).
[Crossref]

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006).
[Crossref]

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

2004 (2)

1999 (1)

Armenise, M. N.

Assefa, S.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Boag, A.

Campanella, C. E.

Campbell, K.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Canciamilla, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Capmany, J.

Cardenas, J.

Ciminelli, C.

Cooper, M. L.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Culshaw, B.

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

De La Rue, R. M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Dell’Olio, F.

DeRose, G. A.

Digonnet, M. J. F.

Domenech, J. D.

Donati, S.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

Fainman, Y.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Fan, S.

Ferrari, C.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Green, W. M. J.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Groisman, A.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Gupta, G.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Huang, Y.

Kaston, Z. A.

Kuo, Y.-H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Lee, R. K.

Levy, U.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Lipson, M.

Liu, H.-C.

Luo, L.-W.

Melloni, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Merlo, S.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

Mookherjea, S.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004).
[Crossref] [PubMed]

Morichetti, F.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Munoz, P.

Muriel, M. A.

Norgia, M.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

O’Boyle, M.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

O’Brien, J. D.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Paloczi, G. T.

Poitras, C.

Poon, J. K. S.

Rooks, M.

Ruffin, P. B.

P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Scherer, A.

Scheuer, J.

Schneider, M. A.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Search, C. P.

Sekaric, L.

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

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Sorel, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Sorrentino, C.

Stapleton, A.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Steier, W. H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Steinberg, B. Z.

Tazawa, H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Terrel, M. A.

Toland, J. R. E.

Torregiani, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

Vlasov, Y.

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

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

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Vlasov, Y. A.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Wiederhecker, G. S.

Xia, F.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

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

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Xu, Y.

Yariv, A.

Zhu, L.

Adv. Opt. Photon. (1)

App. Opt. (1)

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

App. Phys. Lett. (2)

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. (1)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. 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, 104008 (2010).
[Crossref]

J. Opt. Soc. Am. B (2)

Meas. Sci. Technol. (1)

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

Nat. Photon. (1)

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

Opt. Exp. (1)

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006).
[Crossref]

Other (2)

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).

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Figures (4)

Fig. 1
Fig. 1 (a) Illustration of micro-resonator CROW gyroscope on planar substrate connected to a 3dB coupler with periodic modulation of evanescent coupling between resonators. (b) Schematic diagram of proposed modulation scheme for evanescent couplings κj. Note that the green and red discs represent the strength of the evanescent coupling between resonators, which alternates between weak (green) and strong (red) coupling or vice versa symmetrically about the center resonator.

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Fig. 2
Fig. 2 (a) Transmission spectrum as a function of Sagnac phase shift per resonator ϕS for the alternating symmetric coupling modulation with κα = 0.1 and κβ = 0.01 (red line). Notice the single transmission resonance at 0 phase. For comparison the transmission through CROWs with uniform coupling constants κ = 0.01 (blue line) and κ = 0.1 (green line) are also shown. In all plots N = 13. (b) Closeup of central transmission resonance for κα = 0.01 as a function of ϕS and κβ for N = 9 rings. One can see clearly the narrowing of the resonance for increasing |κα – κβ|.

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Fig. 3
Fig. 3 (a) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for various coupling modulation strengths with weak coupling at the edges, κα ≪ κβ. Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In this figure Q int − 1 = 0. (b) The same as part (a) except that coupling is strong at the edges, κα ≫ κβ.

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Fig. 4
Fig. 4 (a) Effect of finite Qint on the transmission resonance of the alternating symmetric coupling modulated CROW for κα = 0.01 and κβ = 0.7 for N = 11 resonators. (b) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for finite Qint and coupling constants modulations κα = 0.01 and κβ = 0.3 (corresponding to Fig. 3(a)) and κα = 0.3 and κβ = 0.01 (corresponding to Fig. 3(b)). Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In both figures the resonators have radii R = 25μm and index of refraction n = 1.6

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Equations (9)

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ϕ S = 4 π ω Ω R 2 c 2
U in = i κ 1 ( exp [ i ϕ 1 ] − 1 − κ 1 exp [ i ϕ 1 ] 1 − κ 1 exp [ − i ϕ 1 ] − exp [ − i ϕ 1 ] )
U out = − i κ N + 1 ( exp [ i ϕ N ] − 1 − κ N + 1 exp [ − i ϕ N ] 1 − κ N + 1 exp [ i ϕ N ] − exp [ − i ϕ N ] ) ( i 1 − κ N κ N i κ N i κ N − i 1 − κ N κ N )
U R H M ( j ) = ( i 1 − κ j κ j exp [ i ϕ j ] − i κ j exp [ i ϕ j ] i κ j exp [ − i ϕ j ] − i 1 − κ j κ j exp [ − i ϕ j ] ) ,
U L H M ( j ) = ( − i 1 − κ j κ j exp [ − i ϕ j ] i κ j exp [ − i ϕ j ] − i κ j exp [ i ϕ j ] i 1 − κ j κ j exp [ i ϕ j ] )
T C C W = U out ∏ l = 1 ( N − 3 ) / 2 U L H M ( 2 l + 2 ) U R H M ( 2 l + 1 ) U L H M ( 2 ) U i n
= ( T 11 T 12 T 21 T 22 )
κ j = { κ α j = 1 , 3 , 5 , … ( N ± 1 ) / 2 κ β j = 2 , 4 , 6 , … ( N ∓ 1 ) / 2 κ α j = 2 + ( N ∓ 1 ) / 2 , 4 + ( N ∓ 1 ) / 2 , … , N + 1 κ β j = 2 + ( N ± ) 1 / 2 , 4 + ( N ± 1 ) / 2 , … N
Ω min = λ c 8 π A eff 2 h c Δ f λ η P opt

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