Abstract

Due to its strong piezoelectric effect and photo-elastic property, lithium niobate is widely used for acousto-optical applications. However, conventional bulk lithium niobate waveguide devices exhibit a large footprint and limited light–sound interaction resulting from the weak guiding of light. Here, we report the first acousto-optical modulators with surface acoustic wave generation, phononic cavity, and low-loss photonic waveguide devices monolithically integrated on a 500 nm thick film of lithium niobate on an insulator. Modulation efficiency was optimized by properly arranging the propagation directions of surface acoustic waves and optical guided modes. The effective photo-elastic coefficient extracted by comparing the first and third harmonic modulation signals from an on-chip Mach–Zehnder interferometer indicates the excellent acousto-optical properties of lithium niobate are preserved in the thin film implementation. Such material property finding is of crucial importance in designing various types of acousto-optical devices. Much stronger amplitude modulation was achieved in a high Q (>300,000) optical resonator due to the higher optical sensitivity. Our results pave the path for developing novel acousto-optical devices using thin film lithium niobate.

© 2019 Chinese Laser Press

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References

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2019 (2)

L. Cai, A. Mahmoud, and G. Piazza, “Low-loss waveguides on Y-cut thin film lithium niobate: towards acousto-optic applications,” Opt. Express 27, 9794–9802 (2019).
[Crossref]

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

2018 (2)

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

M. Mahmoud, A. Mahmoud, L. Cai, M. Khan, T. Mukherjee, J. Bain, and G. Piazza, “Novel on chip rotation detection based on the acousto-optic effect in surface acoustic wave gyroscopes,” Opt. Express 26, 25060–25075 (2018).
[Crossref]

2017 (1)

A. Rao and S. Fathpour, “Compact lithium niobate electrooptic modulators,” IEEE J. Sel. Top. Quantum Electron. 24, 3400114 (2017).
[Crossref]

2016 (2)

W. C. Jiang and Q. Lin, “Chip-scale cavity optomechanics in lithium niobate,” Sci. Rep. 6, 36920 (2016).
[Crossref]

K. Fang, M. M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
[Crossref]

2015 (1)

N. Dostart, S. Kim, and G. Bahl, “Giant gain enhancement in surface-confined resonant stimulated Brillouin scattering,” Laser Photon. Rev. 9, 689–705 (2015).
[Crossref]

2014 (4)

S. Tadesse and M. Li, “Sub-optical wavelength acoustic wave modulation of integrated photonic resonators at microwave frequencies,” Nat. Commun. 5, 5402 (2014).
[Crossref]

K. C. Balram, M. Davanço, J. Y. Lim, J. D. Song, and K. Srinivasan, “Moving boundary and photoelastic coupling in GaAs optomechanical resonators,” Optica 1, 414–420 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

H. Jin, F. M. Liu, P. Xu, J. L. Xia, M. L. Zhong, Y. Yuan, J. W. Zhou, Y. X. Gong, W. Wang, and S. N. Zhu, “On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits,” Phys. Rev. Lett. 113, 113601 (2014).
[Crossref]

2013 (3)

2012 (1)

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[Crossref]

2010 (2)

2006 (1)

M. M. de Lima, M. Beck, R. Hey, and P. V. Santos, “Compact Mach-Zehnder acousto-optic modulator,” Appl. Phys. Lett. 89, 121104 (2006).
[Crossref]

2002 (3)

2001 (1)

2000 (1)

1994 (1)

1993 (1)

M. Itano, F. W. Kern, M. Miyashita, and T. Ohmi, “Particle removal from silicon wafer surface in wet cleaning process,” IEEE Trans. Semicond. Manuf. 6, 258–267 (1993).
[Crossref]

1991 (1)

1988 (1)

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[Crossref]

1985 (1)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[Crossref]

1968 (1)

J. J. Campbell and W. R. Jones, “A method for estimating optimal crystal cuts and propagation direction for excitation of piezoelectric surface waves,” IEEE Trans. Sonics Ultrason. 15, 209–217 (1968).
[Crossref]

1965 (1)

R. M. White and F. W. Voltmer, “Direct piezoelectric coupling to surface acoustic waves,” Appl. Phys. Lett. 7, 314–316 (1965).
[Crossref]

Absil, P.

Ansari, S.

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

Armenise, M. N.

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Bahl, G.

N. Dostart, S. Kim, and G. Bahl, “Giant gain enhancement in surface-confined resonant stimulated Brillouin scattering,” Laser Photon. Rev. 9, 689–705 (2015).
[Crossref]

Bain, J.

Balram, K. C.

Bava, G. P.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[Crossref]

Beck, M.

M. M. de Lima, M. Beck, R. Hey, and P. V. Santos, “Compact Mach-Zehnder acousto-optic modulator,” Appl. Phys. Lett. 89, 121104 (2006).
[Crossref]

Benchabane, S.

N. Courjal, S. Benchabane, J. Dahdah, G. Ulliac, Y. Gruson, and V. Laude, “Acousto-optically tunable lithium niobate photonic crystal,” Appl. Phys. Lett. 96, 131103 (2010).
[Crossref]

Berg, N. J.

N. J. Berg and J. M. Pellegrino, Acousto-Optic Signal Processing: Theory and Implementation, 2nd ed. (Marcel Dekker, 1996).

Bertrand, M.

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

Biermann, K.

Bogaerts, W.

Bortz, M. L.

Buscaino, B.

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

Cai, L.

Campbell, J. J.

J. J. Campbell and W. R. Jones, “A method for estimating optimal crystal cuts and propagation direction for excitation of piezoelectric surface waves,” IEEE Trans. Sonics Ultrason. 15, 209–217 (1968).
[Crossref]

Cantarero, A.

Chandrasekhar, S.

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

Chen, X.

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

Courjal, N.

N. Courjal, S. Benchabane, J. Dahdah, G. Ulliac, Y. Gruson, and V. Laude, “Acousto-optically tunable lithium niobate photonic crystal,” Appl. Phys. Lett. 96, 131103 (2010).
[Crossref]

Crespo-Poveda, A.

Dahdah, J.

N. Courjal, S. Benchabane, J. Dahdah, G. Ulliac, Y. Gruson, and V. Laude, “Acousto-optically tunable lithium niobate photonic crystal,” Appl. Phys. Lett. 96, 131103 (2010).
[Crossref]

Dalton, L. R.

Davanço, M.

de Lima, M. M.

Dostart, N.

N. Dostart, S. Kim, and G. Bahl, “Giant gain enhancement in surface-confined resonant stimulated Brillouin scattering,” Laser Photon. Rev. 9, 689–705 (2015).
[Crossref]

Eggleton, B.

Fang, K.

K. Fang, M. M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
[Crossref]

Fathpour, S.

A. Rao and S. Fathpour, “Compact lithium niobate electrooptic modulators,” IEEE J. Sel. Top. Quantum Electron. 24, 3400114 (2017).
[Crossref]

Fejer, M. M.

Gargallo, B.

Gaylord, T. K.

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[Crossref]

Gnewuch, H.

Gong, S.

S. Gong and G. Piazza, “Design and analysis of lithium-niobate-based high electromechanical coupling RF-MEMS resonators for wideband filtering,” IEEE Trans. Microw. Theory Tech. 61, 403–414 (2013).
[Crossref]

Gong, Y. X.

H. Jin, F. M. Liu, P. Xu, J. L. Xia, M. L. Zhong, Y. Yuan, J. W. Zhou, Y. X. Gong, W. Wang, and S. N. Zhu, “On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits,” Phys. Rev. Lett. 113, 113601 (2014).
[Crossref]

Gruson, Y.

N. Courjal, S. Benchabane, J. Dahdah, G. Ulliac, Y. Gruson, and V. Laude, “Acousto-optically tunable lithium niobate photonic crystal,” Appl. Phys. Lett. 96, 131103 (2010).
[Crossref]

Günter, P.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[Crossref]

Hashimoto, K.

K. Hashimoto, Surface Acoustic Wave Devices in Telecommunications Modelling and Simulation (Springer, 1990).

Hey, R.

Hu, H.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[Crossref]

Huang, H.

Itano, M.

M. Itano, F. W. Kern, M. Miyashita, and T. Ohmi, “Particle removal from silicon wafer surface in wet cleaning process,” IEEE Trans. Semicond. Manuf. 6, 258–267 (1993).
[Crossref]

Jiang, W. C.

W. C. Jiang and Q. Lin, “Chip-scale cavity optomechanics in lithium niobate,” Sci. Rep. 6, 36920 (2016).
[Crossref]

Jin, H.

H. Jin, F. M. Liu, P. Xu, J. L. Xia, M. L. Zhong, Y. Yuan, J. W. Zhou, Y. X. Gong, W. Wang, and S. N. Zhu, “On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits,” Phys. Rev. Lett. 113, 113601 (2014).
[Crossref]

Jones, W. R.

J. J. Campbell and W. R. Jones, “A method for estimating optimal crystal cuts and propagation direction for excitation of piezoelectric surface waves,” IEEE Trans. Sonics Ultrason. 15, 209–217 (1968).
[Crossref]

Kahn, J.

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

Kern, F. W.

M. Itano, F. W. Kern, M. Miyashita, and T. Ohmi, “Particle removal from silicon wafer surface in wet cleaning process,” IEEE Trans. Semicond. Manuf. 6, 258–267 (1993).
[Crossref]

Khan, M.

Kim, S.

N. Dostart, S. Kim, and G. Bahl, “Giant gain enhancement in surface-confined resonant stimulated Brillouin scattering,” Laser Photon. Rev. 9, 689–705 (2015).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Kurz, J. R.

Laude, V.

N. Courjal, S. Benchabane, J. Dahdah, G. Ulliac, Y. Gruson, and V. Laude, “Acousto-optically tunable lithium niobate photonic crystal,” Appl. Phys. Lett. 96, 131103 (2010).
[Crossref]

Lepage, G.

Li, M.

S. Tadesse and M. Li, “Sub-optical wavelength acoustic wave modulation of integrated photonic resonators at microwave frequencies,” Nat. Commun. 5, 5402 (2014).
[Crossref]

Lim, J. Y.

Lin, Q.

W. C. Jiang and Q. Lin, “Chip-scale cavity optomechanics in lithium niobate,” Sci. Rep. 6, 36920 (2016).
[Crossref]

Liu, F. M.

H. Jin, F. M. Liu, P. Xu, J. L. Xia, M. L. Zhong, Y. Yuan, J. W. Zhou, Y. X. Gong, W. Wang, and S. N. Zhu, “On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits,” Phys. Rev. Lett. 113, 113601 (2014).
[Crossref]

Liu, S.

Loncar, M.

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

C. Wang, M. Zhang, X. Chen, M. Bertrand, S. Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562, 101–104 (2018).
[Crossref]

Luan, X.

K. Fang, M. M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
[Crossref]

Mahmoud, A.

Mahmoud, M.

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Matheny, M. M.

K. Fang, M. M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
[Crossref]

Miyashita, M.

M. Itano, F. W. Kern, M. Miyashita, and T. Ohmi, “Particle removal from silicon wafer surface in wet cleaning process,” IEEE Trans. Semicond. Manuf. 6, 258–267 (1993).
[Crossref]

Montrosset, I.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[Crossref]

Mukherjee, T.

Muñoz, P.

Noviello, G.

Ohmi, T.

M. Itano, F. W. Kern, M. Miyashita, and T. Ohmi, “Particle removal from silicon wafer surface in wet cleaning process,” IEEE Trans. Semicond. Manuf. 6, 258–267 (1993).
[Crossref]

Painter, O.

K. Fang, M. M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
[Crossref]

Pannell, C. N.

Pant, R.

Parameswaran, K. R.

Passaro, V. M. N.

Pellegrino, J. M.

N. J. Berg and J. M. Pellegrino, Acousto-Optic Signal Processing: Theory and Implementation, 2nd ed. (Marcel Dekker, 1996).

Piazza, G.

Poberaj, G.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[Crossref]

Poulton, C.

Qiao, H.

Rabiei, P.

Rao, A.

A. Rao and S. Fathpour, “Compact lithium niobate electrooptic modulators,” IEEE J. Sel. Top. Quantum Electron. 24, 3400114 (2017).
[Crossref]

Reimer, C.

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

Roelkens, G.

Ross, G. W.

Roussev, R. V.

Route, R. K.

Santos, P. V.

Selvaraja, S.

Shams-Ansari, A.

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. Kahn, and M. Loncar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

Smith, P. G. R.

Sohler, W.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[Crossref]

Song, J. D.

Srinivasan, K.

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Figures (11)

Fig. 1.
Fig. 1. Dependence of refractive index change (normalized to its maximum) on θ S and θ O .
Fig. 2.
Fig. 2. (a) Cross-section rendering of the AO modulator. Note that dimensions are not to scale and the actual SAW of this implementation propagates more deeply in the substrate. (b) and (c) Schematic layouts of MZI- and resonator-type AOMs. Light (green arrows) is coupled to/from the chip by grating couplers. An SAW is launched by the split IDT, and its amplitude is enhanced in the cavity formed by the reflectors (acting as acoustic mirrors). Drawings are conceptual renderings and not to scale. The acoustic cavity (black dashed line) is much larger than the MZI and RT resonators ( 1.2    mm × 5.7    mm for MZI and 2.4    mm × 6.2    mm for the RT).
Fig. 3.
Fig. 3. Dependence of the (a) first and (b) second derivative of transmission T with respect to ϕ on a and r .
Fig. 4.
Fig. 4. Microscope pictures of (a) MZI- and (b) resonator-type AOMs. (c) SEM image of the photonic waveguide (tilted view). (d) SEM image of the IDT region (top view).
Fig. 5.
Fig. 5. (a) Measurement setup for characterizing AOM. PC, polarization controller; RF SG, RF signal generator; DUT, device under test; EDFA, erbium-doped fiber amplifier; PD, photodetector; SA, spectrum analyzer. (b) Detected modulation power (normalized to the maximum) of the first harmonic when the frequency of RF SG is swept. The Q factor of the acoustic cavity is 1800. (c) Measured (triangle) and theoretical (solid line) first, second, and third harmonic signals as functions of the square root of the driving power from the RF SG when the frequency is fixed at 111.725 MHz. All are normalized to the maximum first harmonic modulation power of 34    dBm . (d) Ratio of first to third harmonic signals (log scale) as a function of the square root of the input power from RF SG. p eff and a p are extracted to be 0.053 and 0.073 rad/√mW, respectively.
Fig. 6.
Fig. 6. (a) Transmission of the RT resonator for a broad wavelength range. (b) Measured (blue circle) and Lorentzian fit (red) transmission of one of the resonances around 1601.53 nm. Inset: mode profile of the fundamental TE-like mode.
Fig. 7.
Fig. 7. (a) Measured (triangle) and theoretical (solid line) first and seconnd harmonic signals as functions of the square root of the driving power from RF SG. All are normalized to the maximum first harmonic modulation power of 71.4    dBm . The reason for the modulated power being lower in the resonator-type AOM than in the MZI-type AOM is that the EDFA is not used in the measurement of the RT resonator. Inset: transmissions of RT resonator when P e is off (blue), 8 dBm (green), and 15 dBm (red). (b) Measured (green circle) and theoretical (green dash) first harmonic signal as the laser wavelength is swept around the resonance of the RT. The maximum modulation is located at the highest slope of the transmission curve. (c) The gain of the resonator-type AOM relative to MZI-type AOM as a function of the square root of the driving power.
Fig. 8.
Fig. 8. Displacement components (a)  u z and (b)  u y . The scale bar is 5 µm. Strain field S 3 = u z / z and S 2 = u y / y in the (c)  z and (d)  y direction. Both S 3 and S 2 are normalized to the maximum of S 3 . The LN thin film surface is at y = 0 .
Fig. 9.
Fig. 9. Refractive index ellipsoid of LN. x y z are crystalline principle coordinates. k SAW is in the x z plane and rotates by angle θ S with respect to z . θ O is the relative angle between k opt and k SAW .
Fig. 10.
Fig. 10. Primary (blue) and effective (red) photo-elastic coefficient tensor elements (a)  p 21 p 26 and (b)  p 31 p 36 .
Fig. 11.
Fig. 11. (a) Extracted a and r from the transmission spectra of resonator at three power levels ( P e is off, 8 dBm, and 15 dBm). The black line is the linear fitting of a . (b) The first (green) and second (orange) derivative of T with respect to ϕ , respectively.

Equations (26)

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H 1 = F J 1 2 ( | ϕ AO | ) cos 2 δ s , H 2 = F J 2 2 ( | ϕ AO | ) sin 2 δ s , H 3 = F J 3 2 ( | ϕ AO | ) cos 2 δ s ,
H 1 = F ( d T d ϕ ) 2 | ϕ AO | 2 , H 2 = 1 16 F ( d 2 T d ϕ 2 ) 2 | ϕ AO | 4 ,
x 2 ( cos 2 θ S n o 2 + sin 2 θ S n e 2 ) + y 2 n o 2 + z 2 ( sin 2 θ S n o 2 + cos 2 θ S n e 2 ) + 2 x z ( sin θ S cos θ S n o 2 sin θ S cos θ S n e 2 ) = 1 ,
x 2 [ cos 2 θ S n o 2 + sin 2 θ S n e 2 + Δ ( 1 n 2 ) 1 ] + y 2 [ 1 n o 2 + Δ ( 1 n 2 ) 2 ] + z 2 [ sin 2 θ S n o 2 + cos 2 θ S n e 2 + Δ ( 1 n 2 ) 3 ] + 2 y z Δ ( 1 n 2 ) 4 + 2 x z [ sin θ S cos θ S n o 2 sin θ S cos θ S n e 2 + Δ ( 1 n 2 ) 5 ] + 2 x y Δ ( 1 n 2 ) 6 = 1 ,
Δ ( 1 n 2 ) I = p I J eff S J , I , J = 1 , 2 , , 6 .
p I J eff = p I J ( r I k l k ) ( l i e i J ) ( l i ε i j S l j ) ,
p = [ p 11 p 12 p 13 p 14 0 0 p 12 p 11 p 13 p 14 0 0 p 31 p 31 p 33 0 0 0 p 41 p 41 0 p 44 0 0 0 0 0 0 p 44 p 41 0 0 0 0 p 14 p 11 p 12 2 ] , r = [ 0 r 22 r 13 0 r 22 r 13 0 0 r 33 0 r 42 0 r 42 0 0 r 22 0 0 ] , e = [ 0 0 0 0 e 15 2 e 22 e 22 e 22 0 e 15 0 0 e 31 e 31 e 33 0 0 0 ] , ε = [ ε 11 0 0 0 ε 11 0 0 0 ε 13 ] .
a = [ cos θ S 0 sin θ S 0 1 0 sin θ S 0 cos θ S ] , M = [ cos 2 θ S 0 sin 2 θ S 0 sin 2 θ S 0 0 1 0 0 0 0 sin 2 θ S 0 cos 2 θ S 0 0 sin 2 θ S 0 0 0 cos θ S 0 sin θ S sin 2 θ S 2 0 sin 2 θ S 2 0 cos 2 θ S 0 0 0 0 sin θ S 0 cos θ S ] .
p M p M T ; r M r a T ; e a e M T ; ε a ε a T .
Δ ( 1 n 2 ) 3 = p 32 eff S 2 + p 33 eff S 3 + p 34 eff S 4 Δ n = 1 2 n 3 ( p 32 eff S 2 + p 33 eff S 3 + p 34 eff S 4 ) ,
ϕ AO = 2 π n 3 λ opt L L N ( p 32 eff S 2 + p 33 eff S 3 + p 34 eff S 4 ) | E ( y , z ) | 2 d y d z | E ( y , z ) | 2 d y d z .
ϕ AO = 2 π n 3 λ opt L ( p 33 eff Γ AO , 3 | S 3 | p 32 eff Γ AO , 2 | S 2 | ) cos ω S t = 2 π n 3 λ opt L p eff | S 3 | cos ω S t ,
p eff = p 33 eff Γ AO , 3 p 32 eff Γ AO , 2 R .
Γ AO , 3 = L N S 3 ( y , z ) | E ( y , z ) | 2 d y d z | E ( y , z ) | 2 d y d z , Γ AO , 2 = L N S 2 ( y , z ) | E ( y , z ) | 2 d y d z | E ( y , z ) | 2 d y d z .
U = 1 2 S L C L K S K .
E = Q m P ω S .
| S 3 | = 2 R L N total Q m ( 1 | s 11 | 2 ) P e C 33 L c L T L N π f S .
ϕ AO = 2 π n 3 λ opt p eff 2 L R L N total Q m ( 1 | s 11 | 2 ) P e C 33 L c T L N π f S cos ω S t = | ϕ AO | cos ω S t .
T = 1 2 [ 1 + sin ( ϕ AO + δ s ) ] = 1 2 [ 1 + sin ( | ϕ AO | cos ω S t ) cos δ s + cos ( | ϕ AO | cos ω S t ) sin δ s ] ,
T 0 = 1 2 [ 1 + J 0 ( | ϕ AO | ) sin δ s ] , T 1 = J 1 ( | ϕ AO | ) cos δ s 1 st , ω , T 2 = J 2 ( | ϕ AO | ) sin δ s 2 nd , 2 ω , T 3 = J 3 ( | ϕ AO | ) cos δ s 3 rd , 3 ω ,
H 1 = F J 1 2 ( | ϕ AO | ) cos 2 δ s 1 st , ω , H 2 = F J 2 2 ( | ϕ AO | ) sin 2 δ s 2 nd , 2 ω , H 3 = F J 3 2 ( | ϕ AO | ) cos 2 δ s 3 rd , 3 ω ,
F = ( P laser I L G EDFA G PD ) 2 2 R SA .
T = a 2 + r 2 2 a r cos ϕ 1 + a 2 r 2 2 a r ϕ , ϕ = ϕ R T + ϕ AO ,
T ( ϕ ) = T ( ϕ R T ) + 1 1 ! ( d T d ϕ | ϕ = ϕ R T ) ϕ AO + 1 2 ! ( d 2 T d ϕ 2 | ϕ = ϕ R T ) ϕ AO 2 + .
T 1 = ( d T d ϕ | ϕ = ϕ R T ) | ϕ AO | 1 st , ω , T 2 = 1 4 ( d 2 T d ϕ 2 | ϕ = ϕ R T ) | ϕ AO | 2 2 nd , 2 ω .
H 1 = F ( d T d ϕ | ϕ = ϕ R T ) 2 | ϕ AO | 2 1 st , ω , H 2 = F 16 ( d 2 T d ϕ 2 | ϕ = ϕ R T ) 2 | ϕ AO | 4 2 nd , 2 ω .