Abstract

We propose and demonstrate a polarization-sensitive dual-comb spectroscopy (DCS) technique that employs an electro-optic modulator for determining the anisotropic optical responses of materials. This straightforward extension of the typical DCS setup directly provides amplitudes and phases in two mutually orthogonal directions of the electric field of light. Using this method, we determined the optic axis direction and the anisotropy in the complex refractive index of a sample whose optical parameter is well defined. We estimate a birefringence of the sample to be 5.49(55)×10−5 at a comb tooth in the 780 nm region.

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

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References

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2018 (2)

2017 (4)

2016 (3)

2015 (2)

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

2013 (2)

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

2012 (1)

2009 (2)

F. Ferdous, D. E. Leaird, C.-B. Huang, and A. M. Weiner, “Dual-comb electric-field cross-correlation technique for optical arbitrary waveform characterization,” Opt. Lett. 34(24), 3875–3877 (2009).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

2008 (1)

T. Ganz, M. Brehm, H. G. von Ribbeck, D. W. van der Weide, and F. Keilmann, “Vector frequency-comb Fourier-transform spectroscopy for characterizing metamaterials,” New J. Phys. 10(12), 123007 (2008).
[Crossref]

2006 (1)

2004 (1)

2002 (1)

2001 (1)

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “Ultrahigh scanning speed optical coherence tomography using optical frequency comb generator,” Jpn. J. Appl. Phys. 40(8B), L878–L880 (2001).
[Crossref]

1966 (1)

1954 (1)

1952 (1)

1941 (1)

1927 (1)

H. A. Kramers, “La diffusion de la lumiére par les atomes,” Atti Cong. Intern. Fisica (Transactions of Volta Centenary Congress) 2, 545–557 (1927).

1926 (1)

Aoki, M.

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

Asahara, A.

Asahi, T.

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

Baumann, E.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

Bennett, J. M.

J. M. Bennett, Handbook of Optics, 2nd ed. (ed. M. Bass), (McGraw-Hill, 1995), Vol. 1, Chap. 5.

Bernhardt, B.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), Chap. 1.

Brehm, M.

T. Ganz, M. Brehm, H. G. von Ribbeck, D. W. van der Weide, and F. Keilmann, “Vector frequency-comb Fourier-transform spectroscopy for characterizing metamaterials,” New J. Phys. 10(12), 123007 (2008).
[Crossref]

M. Brehm, A. Schliesser, and F. Keilmann, “Spectroscopic near-field microscopy using frequency combs in the mid-infrared,” Opt. Express 14(23), 11222–11233 (2006).
[Crossref]

Chen, L.

Coddington, I.

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
[Crossref]

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

de L. Kronig, R.

Dong, X.

Durán, V.

Ferdous, F.

Forman, M. L.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007), Chap. 3.

Ganz, T.

T. Ganz, M. Brehm, H. G. von Ribbeck, D. W. van der Weide, and F. Keilmann, “Vector frequency-comb Fourier-transform spectroscopy for characterizing metamaterials,” New J. Phys. 10(12), 123007 (2008).
[Crossref]

Giorgetta, F. R.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

Gohle, C.

Guelachvili, G.

Hänsch, T. W.

Hase, E.

E. Hase, T. Minamikawa, T. Mizuno, S. Miyamoto, R. Ichikawa, Y.-D. Hsieh, K. Shibuya, K. Sato, Y. Nakajima, A. Asahara, K. Minoshima, Y. Mizutani, T. Iwata, H. Yamamoto, and T. Yasui, “Scan-less confocal phase imaging based on dual-comb microscopy,” Optica 5(5), 634–643 (2018).
[Crossref]

T. Minamikawa, Y. -D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610 (2017).
[Crossref]

Hiromoto, N.

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

Holzwarth, R.

Hong, F.-L.

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Hosaka, K.

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Hosako, I.

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

Hsieh, Y. -D.

T. Minamikawa, Y. -D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610 (2017).
[Crossref]

Hsieh, Y.-D.

Huang, C.-B.

Ichikawa, R.

Ideguchi, T.

Inaba, H.

K. A. Sumihara, S. Okubo, M. Okano, H. Inaba, and S. Watanabe, “Polarization-sensitive dual-comb spectroscopy,” J. Opt. Soc. Am. B 34(1), 154–159 (2017).
[Crossref]

T. Minamikawa, Y. -D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610 (2017).
[Crossref]

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Iwakuni, K.

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Iwata, T.

Jekrard, H. G.

Jones, R. C.

Kaneoka, Y.

T. Minamikawa, Y. -D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610 (2017).
[Crossref]

Kang, J.

Keilmann, F.

Kofler, J.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

Kondo, K.

A. Asahara, A. Nishiyama, S. Yoshida, K. Kondo, Y. Nakajima, and K. Minoshima, “Dual-comb spectroscopy for rapid characterization of complex optical properties of solids,” Opt. Lett. 41(21), 4971–4974 (2016).
[Crossref]

K. Kondo, A. Asahara, Y. Wang, I. Shoji, and K. Minoshima, “Precise birefringence measurement of anisotropic materials by dual-comb spectroscopy,” in 2017 Conference on Lasers and Electro-Optics Pacific Rim, (Optical Society of America, 2017), paper s1667.

Kourogi, M.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “Ultrahigh scanning speed optical coherence tomography using optical frequency comb generator,” Jpn. J. Appl. Phys. 40(8B), L878–L880 (2001).
[Crossref]

Kramers, H. A.

H. A. Kramers, “La diffusion de la lumiére par les atomes,” Atti Cong. Intern. Fisica (Transactions of Volta Centenary Congress) 2, 545–557 (1927).

Leaird, D. E.

Lee, S.-J.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “Ultrahigh scanning speed optical coherence tomography using optical frequency comb generator,” Jpn. J. Appl. Phys. 40(8B), L878–L880 (2001).
[Crossref]

Lei, Z.

Minamikawa, T.

Minoshima, K.

Miyamoto, S.

Mizuno, T.

Mizutani, Y.

Nakajima, Y.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Newbury, N.

Newbury, N. R.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Nishiyama, A.

Ohtsu, M.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “Ultrahigh scanning speed optical coherence tomography using optical frequency comb generator,” Jpn. J. Appl. Phys. 40(8B), L878–L880 (2001).
[Crossref]

Okano, M.

K. A. Sumihara, S. Okubo, M. Okano, H. Inaba, and S. Watanabe, “Polarization-sensitive dual-comb spectroscopy,” J. Opt. Soc. Am. B 34(1), 154–159 (2017).
[Crossref]

M. Okano and S. Watanabe, “Anisotropic optical response of optically opaque elastomers with conductive fillers as revealed by terahertz polarization spectroscopy,” Sci. Rep. 6(1), 39079 (2016).
[Crossref]

Okubo, S.

T. Minamikawa, Y. -D. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. 8(1), 610 (2017).
[Crossref]

K. A. Sumihara, S. Okubo, M. Okano, H. Inaba, and S. Watanabe, “Polarization-sensitive dual-comb spectroscopy,” J. Opt. Soc. Am. B 34(1), 154–159 (2017).
[Crossref]

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Onae, A.

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Picqué, N.

Ramachandran, G. N.

Ramaseshan, S.

Rieker, G. B.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

Sasada, H.

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F.-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 μm,” Appl. Phys. Express 8(8), 082402 (2015).
[Crossref]

Sato, K.

Schiller, S.

Schliesser, A.

Shibuya, K.

Shoji, I.

K. Kondo, A. Asahara, Y. Wang, I. Shoji, and K. Minoshima, “Precise birefringence measurement of anisotropic materials by dual-comb spectroscopy,” in 2017 Conference on Lasers and Electro-Optics Pacific Rim, (Optical Society of America, 2017), paper s1667.

Sinclair, L. C.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

Steel, W. H.

Sumihara, K. A.

Swann, W.

Swann, W. C.

F. R. Giorgetta, G. B. Rieker, E. Baumann, W. C. Swann, L. C. Sinclair, J. Kofler, I. Coddington, and N. R. Newbury, “Broadband phase spectroscopy over turbulent air paths,” Phys. Rev. Lett. 115(10), 103901 (2015).
[Crossref]

A. M. Zolot, F. R. Giorgetta, E. Baumann, W. C. Swann, I. Coddington, and N. R. Newbury, “Broad-band frequency references in the near-infrared: Accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer 118, 26–39 (2013).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Takeda, M.

S. R. Tripathi, M. Aoki, M. Takeda, T. Asahi, I. Hosako, and N. Hiromoto, “Accurate complex refractive index with standard of ZnTe measured by terahertz time domain spectroscopy,” Jpn. J. Appl. Phys. 52(4R), 042401 (2013).
[Crossref]

Teleanu, E. L.

Torres-Company, V.

Tripathi, S. R.

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Figures (12)

Fig. 1.
Fig. 1. Simple schematic of the EOM-based PS DCS setup. fr, fr - Δfr: repetition rates of the S-comb and L-comb, respectively. P1, P2, P3: polarizers. BS: beam splitter. Q1, Q2: quarter-wave plates. EOM: electro-optic amplitude modulator. fEOM: modulation frequency of the EOM. The degrees in parentheses represent the angle of transmission or fast optic axes of each optical component with respect to the x-axis.
Fig. 2.
Fig. 2. Conceptual diagram of PS DCS data in the frequency domain for fEOM = Δfr /9. Each black line shows an interference signal between two modes of the S-comb and the L-comb with index n. The orange and the green sidebands arise from the modulation of the retardance by the sinusoidal voltage applied to the EOM. The amplitudes (phases) of the signals with frequencies nΔfr ± fEOM and nΔfr ± 3fEOM are indicators of the amplitudes (phases) in the y direction of the mode n of the S-comb, while those with frequencies nΔfr ± 2fEOM and nΔfr ± 4fEOM reflect those in the x direction.
Fig. 3.
Fig. 3. Experimental setup of PS DCS with an EOM on the detector side. fr, fr –Δfr: repetition rates of the S-comb and L-comb, respectively. PPLN: periodically-poled lithium niobate, P1, P2, P3: polarizers. BS1, BS2, BS3: beam splitters. S: Sample. Q1, Q2: quarter-wave plates. fEOM: modulation frequency of EOM on the detector side. BPF: optical bandpass filter. LPF: low-pass filter. PC: personal computer.
Fig. 4.
Fig. 4. (a) Averaged interferogram Uave(t) on a large time scale and (b) that observed in narrow time range. In the latter, we can confirm the signals from the sample-path (earlier) and that from the reference-path (delayed). (c) Normalized amplitude of the Fourier component of US(t) in the optical frequency region. (d) Normalized RF amplitude derived from the Fourier transform of US(t). (e) RF amplitude derived using the additional analysis described in Appendix B.
Fig. 5.
Fig. 5. Experimental results of Stokes parameters of the mode at 385.034024 THz. Solid curves are theoretical curves.
Fig. 6.
Fig. 6. θQWP dependence of θslow. Red symbols represent the experimental data and the black solid line represents the fitting result.
Fig. 7.
Fig. 7. (a) Phase differences between the E-field components with and without sample along the fast (red) and the slow (blue) optic axes. (b) Phase retardance between the slow and the fast optic axes in (a). (c) Transmittance (for the intensity) along the fast (red) and the slow (blue) optic axes.
Fig. 8.
Fig. 8. (a) The real and (b) the imaginary part of the anisotropic complex refractive index of the sample.
Fig. 9.
Fig. 9. (a) N and (b) K of the sample along the fast (red) and slow (blue) axes at 385.034024 THz. (c) The standard deviations of the mean of N and (d) that of K for various θslow along the fast (red) and slow (blue) axes.
Fig. 10.
Fig. 10. Birefringence of the sample for 385.034024 THz measured at various θslow.
Fig. 11.
Fig. 11. Experimental results of mEOM when we set fEOM at (a) Δfr/7, (b) Δfr/9, and (c) Δfr/11.
Fig. 12.
Fig. 12. Voltage dependence of the retardance C induced by the EOM. V0 (Vπ): Value of the voltage applied to the EOM required to induce a retardance of zero (π).

Equations (55)

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E S,x ( t ) = n S E S,x n S exp ( i ϕ S,x n S ) exp [ i 2 π ( f S,ceo + n S f r ) t ] ,
E S,y ( t ) = n S E S,y n S exp ( i ϕ S,y n S ) exp [ i 2 π ( f S,ceo + n S f r ) t ] .
E L,x ( t ) = n L E L,x n L exp ( i ϕ L,x n L ) exp [ i 2 π { f L,ceo + n L ( f r Δ f r ) } t ] .
E S,x ( t ) = n S E S,x n S exp ( i ϕ S,x n S ) exp [ i 2 π ( f low + n S f r ) t ] ,
E S,y ( t ) = n S E S,y n S exp ( i ϕ S,y n S ) exp [ i 2 π ( f low + n S f r ) t ] ,
E L,x ( t ) = n L E L,x n L exp ( i ϕ L,x n L ) exp [ i 2 π { f low + n L ( f r Δ f r ) } t ] .
E x ( t ) = E L,x ( t ) + E S,x ( t ) ,
E y ( t ) = E S,y ( t ) .
M Q1 = 1 2 ( 1 + i 0 0 1 i ) M EOM = ( cos π 4 sin π 4 sin π 4 cos π 4 ) ( e i C 2 0 0 e i C 2 ) ( cos π 4 sin π 4 sin π 4 cos π 4 ) M Q2 = 1 2 ( 1 i 0 0 1 + i ) M P3 = ( 1 0 0 0 )
( E x,det ( t ) E y,det ( t ) ) = M P3 M Q2 M EOM M Q1 ( E x ( t ) E y ( t ) ) .
E x,det ( t ) = cos C 2 { E L,x ( t ) + E S,x ( t ) } sin C 2 E S,y ( t ) .
U ( t ) = cos 2 C 2 { E S,x ( t ) E L,x ( t ) + E L,x ( t ) E S,x ( t ) } cos C 2 sin C 2 { E L,x ( t ) E S,y ( t ) + E S,y ( t ) E L,x ( t ) }
E S,x ( t ) E L,x ( t ) = n S n L E S,x n S E L,x n L exp [ i ( ϕ S,x n S ϕ L,x n L ) ] exp { i 2 π [ ( n S n L ) f r + n L Δ f r ] t } .
0 < ( n S n L ) f r + n L Δ f r < f r 2 .
E S,x ( t ) E L,x ( t ) = n E S,x n E L,x n exp [ i ( ϕ S,x n ϕ L,x n ) ] exp ( i 2 π n Δ f r t ) .
U ( t ) = n { ( 1 + cos C ) E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) sin C E S,y n E L,x n cos ( 2 π n Δ f r t + ϕ S,y n ϕ L,x n ) } .
C ( V ( t ) , f opt ) = π V π ( f opt ) V ( t ) ,
V ( t ) = V pp 2 sin ( 2 π f EOM t + φ EOM ) .
m EOM ( f opt ) = π 2 V pp V π ( f opt ) .
U ( t ) = n [ ( 1 + J 0 ( m EOM ) ) E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) + k = 1 J 2k ( m EOM ) E S,x n E L,x n { cos [ 2 π ( n Δ f r + 2 k f EOM ) t + 2 k φ EOM + ϕ S,x n ϕ L,x n ] . + cos [ 2 π ( n Δ f r 2 k f EOM ) t 2 k φ EOM + ϕ S,x n ϕ L,x n ] } k = 1 J 2k - 1 ( m EOM ) E S,y n E L,x n { sin [ 2 π ( n Δ f r + ( 2 k 1 ) f EOM ) t + ( 2 k 1 ) φ EOM + ϕ S,y n ϕ L,x n ] sin [ 2 π ( n Δ f r ( 2 k 1 ) f EOM ) t ( 2 k 1 ) φ EOM + ϕ S,y n ϕ L,x n ] } ]
A n, ± 2k J 2k ( m EOM ) E S,x n E L,x n A n, ± ( 2k - 1 ) J 2k - 1 ( m EOM ) E S,y n E L,x n Φ n, ± 2k ± 2 k φ EOM + ϕ S,x n ϕ L,x n Φ n, ± ( 2k - 1 ) ± ( 2 k 1 ) φ EOM + ϕ S,y n ϕ L,x n
A n, ± 3 A n, ± 1 = J 3 ( m EOM ) J 1 ( m EOM ) .
( A n,3 A n,1 + A n, - 3 A n, - 1 ) 2 = J 3 ( m EOM ) J 1 ( m EOM ) .
E S,x n E L,x n = A n,2 + A n, - 2 2 J 2 ( m EOM ) ,
E S,y n E L,x n = A n,1 + A n, - 1 2 J 1 ( m EOM ) .
ϕ S,x n ϕ L,x n = Φ n,2 + Φ n, - 2 2
ϕ S,y n ϕ L,x n = Φ n,1 + Φ n, - 1 2
S 1,n w/ = | E S,x w/,n | 2 | E S,y w/,n | 2 = 1 | E L,x n | 2 ( | A n,2 + A n, - 2 2 J 2 ( m EOM ) | 2 | A n,1 + A n, - 1 2 J 1 ( m EOM ) | 2 ) S 2,n w/ = 2 E S,x w/,n E S,y w/,n cos ( ϕ S,x w/,n ϕ S,y w/,n ) = 2 | E L,x n | 2 A n,2 + A n, - 2 2 J 2 ( m EOM ) A n,1 + A n, - 1 2 J 1 ( m EOM ) cos ( Φ n,2 + Φ n, - 2 2 Φ n,1 + Φ n, - 1 2 ) S 3,n w/ = 2 E S,x w/,n E S,y w/,n sin ( ϕ S,x w/,n ϕ S,y w/,n ) = 2 | E L,x n | 2 A n,2 + A n, - 2 2 J 2 ( m EOM ) A n,1 + A n, - 1 2 J 1 ( m EOM ) sin ( Φ n,2 + Φ n, - 2 2 Φ n,1 + Φ n, - 1 2 )
( E S, slow n E L,x n exp ( i ϕ S, slow n ) E S, fast n E L,x n exp ( i ϕ S, fast n ) ) = ( cos θ slow sin θ slow sin θ slow cos θ slow ) ( E S,x n E L,x n exp [ i ( ϕ S,x n ϕ L,x n ) ] E S,y n E L,x n exp [ i ( ϕ S,y n ϕ L,x n ) ] ) .
T slow n = | E S, slow w/,n | 2 | E S, slow w/o,n | 2
T fast n = | E S, fast w/,n | 2 | E S, fast w/o,n | 2
Δ ϕ S, slow n = ϕ S, slow w/,n ϕ S, slow w/o,n ,
Δ ϕ S,fast n = ϕ S, fast w/,n ϕ S, fast w/o,n .
T slow ( fast ) n exp ( i Δ ϕ S, slow ( fast ) n ) = 4 N ^ slow ( fast ) n N air ( N ^ slow ( fast ) n + N air ) 2 exp ( i ( N ^ slow ( fast ) n N air ) d c 2 π f opt n ) ,
S 1,17070 = 1 2 ( 1 + cos 4 ( θ QWP + θ 0 ) ) S 2,17070 = 1 2 sin 4 ( θ QWP + θ 0 ) S 3,17070 = sin 2 ( θ QWP + θ 0 )
A n,3 A n,1 = J 3 ( m EOM ) 2 ( E S,y n E L,x n ) 2 + J 2 ( l - 1 ) ( m EOM ) 2 ( E S,x n + 1 E L,x n + 1 ) 2 + 2 J 3 ( m EOM ) J 2 ( l - 1 ) ( m EOM ) E S,y n E L,x n E S,x n + 1 E L,x n + 1 sin [ ( 2 l + 1 ) φ EOM + ( ϕ S,x n + 1 ϕ L,x n + 1 ϕ S,y n + ϕ L,x n ) ] J 1 ( m EOM ) 2 ( E S,y n E L,x n ) 2 + J 2l ( m EOM ) 2 ( E S,x n + 1 E L,x n + 1 ) 2 + 2 J 1 ( m EOM ) J 2l ( m EOM ) E S,y n E L,x n E S,x n + 1 E L,x n + 1 sin [ ( 2 l + 1 ) φ EOM + ( ϕ S,x n + 1 ϕ L,x n + 1 ϕ S,y n + ϕ L,x n ) ] = J 3 ( m EOM ) J 1 ( m EOM )
A n, - 3 A n, - 1 = J 3 ( m EOM ) 2 ( E S,y n E L,x n ) 2 + J 2 ( l - 1 ) ( m EOM ) 2 ( E S,x n - 1 E L,x n - 1 ) 2 2 J 3 ( m EOM ) J 2 ( l - 1 ) ( m EOM ) E S,y n E L,x n E S,x n - 1 E L,x n - 1 sin [ ( 2 l + 1 ) φ EOM + ( ϕ S,x n - 1 ϕ L,x n - 1 ϕ S,y n + ϕ L,x n ) ] J 1 ( m EOM ) 2 ( E S,y n E L,x n ) 2 + J 2l ( m EOM ) 2 ( E S,x n - 1 E L,x n - 1 ) 2 2 J 1 ( m EOM ) J 2 l ( m EOM ) E S,y n E L,x n E S,x n - 1 E L,x n - 1 sin [ ( 2 l + 1 ) φ EOM + ( ϕ S,x n - 1 ϕ L,x n - 1 ϕ S,y n + ϕ L,x n ) ] = J 3 ( m EOM ) J 1 ( m EOM )
( A n,3 A n,1 + A n, - 3 A n, - 1 ) 2 = J 3 ( m EOM ) J 1 ( m EOM )
C mod ( V ( t ) , f opt ) = π V ( t ) V π ( f opt ) V 0 ( f opt ) π V 0 ( f opt ) V π ( f opt ) V 0 ( f opt ) ,
V ( t ) = V pp 2 sin ( 2 π f EOM t + φ EOM ) + V off
C mod ( t , f opt ) = m EOM,mod ( f opt ) sin ( 2 π f EOM t + φ EOM ) + δ ( f opt ) ,
m EOM,mod ( f opt ) V pp V π ( f opt ) V 0 ( f opt ) π 2 δ ( f opt ) π V off V 0 ( f opt ) V π ( f opt ) V 0 ( f opt )
α 2 k = 1 J 2k ( m EOM, mod ) cos ( 2 π 2 k f EOM t + 2 k φ EOM ) ,
β 2 k = 1 J 2k - 1 ( m EOM,mod ) sin [ 2 π ( 2 k 1 ) f EOM t + ( 2 k 1 ) φ EOM ] .
cos C ( t ) = ( J 0 ( m EOM,mod ) + α ) cos δ β sin δ sin C ( t ) = β cos δ + ( J 0 ( m EOM,mod ) + α ) sin δ
U ( t ) n { ( 1 + J 0 ( m EOM,mod ) cos δ ) E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) J 0 ( m EOM,mod ) sin δ E S,y n E L,x n cos ( 2 π n Δ f r t + ϕ S,y n ϕ L,x n ) + ( α cos δ β sin δ ) E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) ( β cos δ + α sin δ ) E S,y n E L,x n cos ( 2 π n Δ f r t + ϕ S,y n ϕ L,x n ) } .
U ( t ) = n { ( α cos δ β sin δ ) E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) ( β cos δ + α sin δ ) E S,y n E L,x n cos ( 2 π n Δ f r t + ϕ S,y n ϕ L,x n ) } .
( E S,x n exp ( i ϕ S,x n ) E S,y n exp ( i ϕ S,y n ) ) = R 1 ( δ ) ( E S,x n exp ( i ϕ S,x n ) E S,y n exp ( i ϕ S,y n ) ) ,
R ( δ ) = ( cos δ sin δ sin δ cos δ ) .
U ( t ) = n { α E S,x n E L,x n cos ( 2 π n Δ f r t + ϕ S,x n ϕ L,x n ) β E S,y n E L,x n cos ( 2 π n Δ f r t + ϕ S,y n ϕ L,x n ) } .
U ( t ) = n ( k = 1 J 2k ( m EOM,mod ) E S,x n E L,x n { cos [ 2 π ( n Δ f r + 2 k f EOM ) t + 2 k φ EOM + ϕ S,x n ϕ L,x n ] + cos [ 2 π ( n Δ f r 2 k f EOM ) t 2 k φ EOM + ϕ S,x n ϕ L,x n ] } k = 1 J 2k - 1 ( m EOM,mod ) E S,y n E L,x n { sin [ 2 π ( n Δ f r + ( 2 k 1 ) f EOM ) t + ( 2 k 1 ) φ EOM + ϕ S,y n ϕ L,x n ] sin [ 2 π ( n Δ f r ( 2 k 1 ) f EOM ) t ( 2 k 1 ) φ EOM + ϕ S,y n ϕ L,x n ] } )
( E S,x n exp ( i ϕ S,x n ) E S,y n exp ( i ϕ S,x n ) ) = R ( δ ) ( E S,x n exp ( i ϕ S,x n ) E S,y n exp ( i ϕ S,y n ) ) .
U ( t ) = n k = 1 [ J 2k ( m EOM,mod ) cos δ E S,x n E L,x n { cos [ 2 π ( n Δ f r + 2 k f EOM ) t + ϕ S,x n ϕ L,x n + 2 k φ EOM ] + cos [ 2 π ( n Δ f r 2 k f EOM ) t + ϕ S,x n ϕ L,x n 2 k φ EOM ] } J 2k - 1 ( m EOM,mod ) sin δ E S,x n E L,x n { sin [ 2 π ( n Δ f r + ( 2 k 1 ) f EOM ) t + ϕ S,x n ϕ L,x n + ( 2 k 1 ) φ EOM ] sin [ 2 π ( n Δ f r ( 2 k 1 ) f EOM ) t + ϕ S,x n ϕ L,x n ( 2 k 1 ) φ EOM ] } ]
A n,1 + A n, - 1 A n,2 + A n, - 2 = J 1 ( m EOM,mod ) sin δ J 2 ( m EOM,mod ) cos δ
δ = ta n 1 ( A n,1 + A n, - 1 A n,2 + A n, - 2 J 2 ( m EOM,mod ) J 1 ( m EOM,mod ) ) .