Abstract

In this work, we investigate the Fourier characteristics of wavelength-scanned optical spectrum of low-finesse Fabry-Pérot (FP) acoustic sensor both theoretically and experimentally. The wavelength scanning will transform the time-domain acoustic signal into phase modulation loaded on the FP sensor spectrum distributed along wavelength. Therefore the interference spectrum can be regarded as carrier signal in the wavelength domain. From this perspective, it is intelligible that the phase modulation loaded on the spectrum (carrier signal) will introduce sidebands in Fourier domain. The spatial frequency and phase of sideband components contain unique information of both acoustic signal and the corresponding sensor. These conclusions are experimentally proved by single sensor head as well as two parallel sensors. The Fourier characteristics of sideband components can be utilized to recognize and distinguish acoustic signals received by different sensors, indicating that it has potential applications in multiplexed FP sensor array and source localization, and so forth.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Z. Gong, K. Chen, X. Zhou, Y. Yang, Z. Zhao, H. Zou, and Q. Yu, “High-sensitivity Fabry-Perot interferometric acoustic sensor for low-frequency acoustic pressure detections,” J. Lightwave Technol. 35(24), 5276–5279 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  19. X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).
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    [Crossref]
  21. A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
    [Crossref]
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    [Crossref]

2018 (1)

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

2017 (4)

2016 (5)

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

J. Xia, S. Xiong, F. Wang, and H. Luo, “Wavelength-switched phase interrogator for extrinsic Fabry-Perot interferometric sensors,” Opt. Lett. 41(13), 3082–3085 (2016).
[Crossref] [PubMed]

D. Tosi, “Simultaneous detection of multiple fiber-optic Fabry–Perot interferometry sensors with cepstrum-division multiplexing,” J. Lightwave Technol. 34(15), 3622–3627 (2016).
[Crossref]

Z. Yu and A. Wang, “Fast demodulation algorithm for multiplexed low-finesse Fabry-Pérot interferometers,” J. Lightwave Technol. 34(3), 1015–1019 (2016).
[Crossref]

2015 (3)

2014 (3)

2013 (2)

C. Ma and A. Wang, “Signal processing of white-light interferometric low-finesse fiber-optic Fabry-Perot sensors,” Appl. Opt. 52(2), 127–138 (2013).
[Crossref] [PubMed]

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

2008 (2)

Y. Jiang, “Fourier‐transform phase comparator for the measurement of extrinsic Fabry-Perot interferometric sensors,” Microw. Opt. Technol. Lett. 50(10), 2621–2625 (2008).
[Crossref]

Y. Jiang and C. Tang, “Fourier transform white-light interferometry based spatial frequency-division multiplexing of extrinsic Fabry-Pérot interferometric sensors,” Rev. Sci. Instrum. 79(10), 106105 (2008).
[Crossref] [PubMed]

1982 (1)

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Chen, K.

Dandridge, A.

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Digonnet, M.

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

Fan, S.

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Fang, D.

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

Fu, X.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

Giallorenzi, T.

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Gong, K.

Gong, Z.

Ho, H.

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Hu, Y.

Hu, Z.

Hui, R.

Jan, C.

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

Jiang, J.

Jiang, X.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

Jiang, Y.

Y. Jiang, “Fourier‐transform phase comparator for the measurement of extrinsic Fabry-Perot interferometric sensors,” Microw. Opt. Technol. Lett. 50(10), 2621–2625 (2008).
[Crossref]

Y. Jiang and C. Tang, “Fourier transform white-light interferometry based spatial frequency-division multiplexing of extrinsic Fabry-Pérot interferometric sensors,” Rev. Sci. Instrum. 79(10), 106105 (2008).
[Crossref] [PubMed]

Jin, W.

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Jo, W.

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

W. Jo, O. Kilic, and J. F. Michel, “Highly sensitive phase-front-modulation fiber acoustic sensor,” J. Lightwave Technol. 33(20), 4377–4383 (2015).
[Crossref]

Kilic, O.

Li, H.

Liao, H.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

Liu, D.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

Liu, K.

Liu, L.

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

Liu, T.

Lu, P.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

Luo, H.

Ma, C.

Ma, J.

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Mao, X.

Michel, J. F.

Ni, W.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

Pan, Y.

Qin, Z.

Shao, Z.

Shi, J.

Solgaard, O.

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

Sun, A.

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

Tang, C.

Y. Jiang and C. Tang, “Fourier transform white-light interferometry based spatial frequency-division multiplexing of extrinsic Fabry-Pérot interferometric sensors,” Rev. Sci. Instrum. 79(10), 106105 (2008).
[Crossref] [PubMed]

Tosi, D.

Tveten, A.

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Wang, A.

Wang, F.

Wang, S.

Wang, W.

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

Wang, X.

Wu, Z.

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

Xia, J.

Xie, J.

Xiong, S.

Xu, F.

Xuan, H.

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Yang, Y.

Z. Gong, K. Chen, X. Zhou, Y. Yang, Z. Zhao, H. Zou, and Q. Yu, “High-sensitivity Fabry-Perot interferometric acoustic sensor for low-frequency acoustic pressure detections,” J. Lightwave Technol. 35(24), 5276–5279 (2017).
[Crossref]

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

Yin, J.

Yu, B.

Yu, Q.

Yu, Z.

Z. Yu and A. Wang, “Fast demodulation algorithm for multiplexed low-finesse Fabry-Pérot interferometers,” J. Lightwave Technol. 34(3), 1015–1019 (2016).
[Crossref]

Z. Yu and A. Wang, “Fast white light interferometry demodulation algorithm for low-finesse Fabry–Pérot sensors,” IEEE Photonics Technol. Lett. 27(8), 817–820 (2015).
[Crossref]

Yuan, S.

Zhang, J.

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

X. Fu, P. Lu, W. Ni, H. Liao, S. Wang, D. Liu, and J. Zhang, “Spectrum interrogation of fiber acoustic sensor based on self-fitting and differential method,” Opt. Express 25(4), 4429–4437 (2017).
[Crossref] [PubMed]

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

Zhao, Z.

Zheng, P.

Zhou, X.

Zou, H.

Zou, S.

Appl. Opt. (1)

Chin. Opt. Lett. (1)

IEEE J. Quantum Electron. (1)

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

IEEE Photonics J. (2)

X. Fu, P. Lu, W. Ni, H. Liao, X. Jiang, D. Liu, and J. Zhang, “Phase interrogation of diaphragm-based optical fiber acoustic sensor assisted by wavelength-scanned spectral coding,” IEEE Photonics J. 10(3), 1–11 (2018).

H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, “Phase demodulation of short-cavity Fabry–Perot interferometric acoustic sensors with two wavelengths,” IEEE Photonics J. 9(2), 1–9 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (3)

J. Ma, H. Xuan, H. Ho, W. Jin, Y. Yang, and S. Fan, “Fiber-optic Fabry–Pérot acoustic sensor with multilayer graphene diaphragm,” IEEE Photonics Technol. Lett. 25(10), 932–935 (2013).
[Crossref]

C. Jan, W. Jo, M. Digonnet, and O. Solgaard, “Photonic-crystal-based fiber hydrophone with sub-100 μPa/√Hz pressure resolution,” IEEE Photonics Technol. Lett. 28(2), 123–126 (2016).
[Crossref]

Z. Yu and A. Wang, “Fast white light interferometry demodulation algorithm for low-finesse Fabry–Pérot sensors,” IEEE Photonics Technol. Lett. 27(8), 817–820 (2015).
[Crossref]

IEEE Sens. J. (1)

A. Sun, Z. Wu, D. Fang, J. Zhang, and W. Wang, “Multimode interference-based fiber-optic ultrasonic sensor for non-contact displacement measurement,” IEEE Sens. J. 16(14), 5632–5635 (2016).
[Crossref]

J. Lightwave Technol. (6)

Microw. Opt. Technol. Lett. (1)

Y. Jiang, “Fourier‐transform phase comparator for the measurement of extrinsic Fabry-Perot interferometric sensors,” Microw. Opt. Technol. Lett. 50(10), 2621–2625 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Rev. Sci. Instrum. (1)

Y. Jiang and C. Tang, “Fourier transform white-light interferometry based spatial frequency-division multiplexing of extrinsic Fabry-Pérot interferometric sensors,” Rev. Sci. Instrum. 79(10), 106105 (2008).
[Crossref] [PubMed]

Other (1)

S. Hu, B. Dong, K. Yu, J. Zhou, and L. Wang, “A hydrophone based on high-birefringence fiber loop mirror,” 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Smart Structures and Materials in Manufacturing and Testing (International Society for Optics and Photonics, 2010), paper 76590W.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Experimental setup to verify the Fourier characteristics using single sensor head.
Fig. 2
Fig. 2 (a) WSOS of the sensor under exposure to 50 Hz acoustic wave; (b) Spatial frequency spectrum of the WSOS.
Fig. 3
Fig. 3 (a) FFT spectra of the WSOSs when acoustic wave with different frequency is applied; (b) Relationship between the acoustic frequency and spatial frequency.
Fig. 4
Fig. 4 Sensor spatial frequency and sideband interval under different acoustic frequency.
Fig. 5
Fig. 5 FFT spectra of the WSOSs with different scanning speed while the same 100 Hz acoustic signal is applied.
Fig. 6
Fig. 6 (a) FFT spectra of the WSOS under 200 Hz signal with different sound pressure; (b) Relationship between sound pressure with the intensities of spatial frequency component and sideband components.
Fig. 7
Fig. 7 Experimental setup using two parallel sensors.
Fig. 8
Fig. 8 (a) Optical spectra of the two individual sensors with different cavity length. (b) Composite spectrum of the two sensors. (c) FFT spectrum when 50 Hz signal is applied to both sensors. (d) FFT vectors at spatial frequencies corresponding to the 2 sensors. (e)FFT vectors of the sideband components. (f) Phase map for sideband 1 recognition. (g) Phase map for sideband 2 recognition.

Tables (1)

Tables Icon

Table 1 Phase Condition Related to the Symbol of Bessel Function of C

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

{ R(λ)=A+Bcos(βλ+x(λ)+φ) x(λ)=Csin( ω 0 λ+α) C= 4πΔ L m / λ 0 , ω 0 = ω a /V , α= α a -( ω a λ 0 /V )
F(ω)=F[R(λ)]=F[A]+BF[cos(βλ+φ+x(λ))] =F[A]+ B 2π F[cos(βλ+φ)]F[cos(x(λ))] B 2π F[sin(βλ+φ)]F[sin(x(λ))]
{ cos(x(λ))= J 0 (C)+2 J 2 (C)cos(2 ω 0 λ+2α) sin(x(λ))=2 J 1 (C)sin( ω 0 λ+α)+2 J 3 (C)sin(3 ω 0 λ+3α)
{ F[A]=2πAδ(ω) F[cos(βλ+φ)]=π(cosφjsinφ)δ(ω+β)+π(cosφ+jsinφ)δ(ωβ) F[sin(βλ+φ)]=π(sinφ+jcosφ)δ(ω+β)+π(sinφjcosφ)δ(ωβ) F[cos(x(λ))]=2π J 0 (C)δ(ω)+2π J 2 (C)cos(2α)[δ(ω+2 ω 0 )+δ(ω2 ω 0 )] j2π J 2 (C)sin(2α)[δ(ω+2 ω 0 )δ(ω2 ω 0 )] F[sin(x(λ))]=2π J 1 (C){ sinα[δ(ω+ ω 0 )+δ(ω ω 0 )]+jcosα[δ(ω+ ω 0 )δ(ω ω 0 )] } +2π J 3 (C){ sin3α[δ(ω+3 ω 0 )+δ(ω3 ω 0 )]+jcos3α[δ(ω+3 ω 0 )δ(ω3 ω 0 )] }
{ F(ω)= F DC + F S + F L + F R + F 2nd + F 3rd F DC =2πAδ(ω) F S =πB J 0 (C)(cosφ+jsinφ)δ(ωβ) F L =πB J 1 (C)[ cos(αφ)+jsin(αφ) ]δ(ω ω 0 +β) F R =πB J 1 (C)[ cos(α+φ)+jsin(α+φ) ]δ(ω ω 0 β)
{ V S =πB J 0 (C)(cosφ+jsinφ) V L =πB J 1 (C)[ cos(αφ)+jsin(αφ) ] V R =πB J 1 (C)[ cos(α+φ)+jsin(α+φ) ]
{ cos φ S = πB J 0 (C)cosφ πB J 0 (C) =cosφ, sin φ S = πB J 0 (C)sinφ πB J 0 (C) =sinφ cos θ L = πB J 1 (C)cos(αφ) πB J 1 (C) =cos(αφ), sin θ L = πB J 1 (C)sin(αφ) πB J 1 (C) =sin(αφ) cos θ R = πB J 1 (C)cos(α+φ) πB J 1 (C) =cos(α+φ), sin θ R = πB J 1 (C)sin(α+φ) πB J 1 (C) =sin(α+φ)
{ sin( θ R θ L )=sin(2 φ S +π) cos( θ R θ L )=cos(2 φ S +π)