Abstract

In this paper, we demonstrated an improved laser self-mixing grating interferometer (SMGI) with auto-collimation design which can avoid the disturbance from the light feedback of the zero-order diffraction beam. In order to obtain higher optical subdivision, SMGI with multiple-diffraction is implemented. Both theoretical analysis and experimental work show that the proposed system for displacement measurement can achieve high sensitivity and low measurement uncertainty. Using the proposed system, different forms of micro-displacement signals applied on the target (grating) have been reconstructed with accuracy of a few nanometers. The work presented in this paper provides a good way to achieve robust and high precision measurement with compact system configuration.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).
  2. A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
  3. S. Donati and M. Norgia, “Self-mixing interferometry for biomedical signals sensing,” IEEE J. Sel. Top. Quantum Electron. 20(2), 104–111 (2014).
  4. S. Zhang and W. Holzapfel, Orthogonal Polarization in Lasers: Physical Phenomena and Engineering Applications (John Wiley and Sons, 2013).
  5. S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).
  6. K. Zhu, B. Guo, Y. Lu, S. Zhang, and Y. Tan, “Single-spot two-dimensional displacement measurement based on self-mixing interferometry,” Optica 4(7), 729–735 (2017).
  7. M. T. Fathi and S. Donati, “Thickness measurement of transparent plates by a self-mixing interferometer,” Opt. Lett. 35(11), 1844–1846 (2010).
    [PubMed]
  8. Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
    [PubMed]
  9. S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).
  10. Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).
  11. Y. Fan, Y. Yu, J. Xi, and J. F. Chicharo, “Improving the measurement performance for a self-mixing interferometry-based displacement sensing system,” Appl. Opt. 50(26), 5064–5072 (2011).
    [PubMed]
  12. Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009).
    [PubMed]
  13. K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
    [PubMed]
  14. D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).
  15. D. Guo and M. Wang, “Note: Design of a laser feedback interferometer with double diffraction system,” Rev. Sci. Instrum. 86(9), 096111 (2015).
    [PubMed]
  16. J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).
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    [PubMed]
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  20. W. Xia, M. Wang, Z. Yang, W. Guo, H. Hao, and D. Guo, “High-accuracy sinusoidal phase-modulating self-mixing interferometer using an electro-optic modulator: development and evaluation,” Appl. Opt. 52(4), B52–B59 (2013).
    [PubMed]

2017 (1)

2016 (3)

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

2015 (2)

S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).

D. Guo and M. Wang, “Note: Design of a laser feedback interferometer with double diffraction system,” Rev. Sci. Instrum. 86(9), 096111 (2015).
[PubMed]

2014 (1)

S. Donati and M. Norgia, “Self-mixing interferometry for biomedical signals sensing,” IEEE J. Sel. Top. Quantum Electron. 20(2), 104–111 (2014).

2013 (3)

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

W. Xia, M. Wang, Z. Yang, W. Guo, H. Hao, and D. Guo, “High-accuracy sinusoidal phase-modulating self-mixing interferometer using an electro-optic modulator: development and evaluation,” Appl. Opt. 52(4), B52–B59 (2013).
[PubMed]

2011 (2)

2010 (2)

M. T. Fathi and S. Donati, “Thickness measurement of transparent plates by a self-mixing interferometer,” Opt. Lett. 35(11), 1844–1846 (2010).
[PubMed]

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

2009 (1)

2008 (1)

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

2007 (1)

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

2001 (1)

T. Bosch and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40(1), 20–27 (2001).

Arai, Y.

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

Bosch, T.

T. Bosch and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40(1), 20–27 (2001).

Chen, H. Y.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

Cheng, F.

Chicharo, J. F.

Donati, S.

S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).

S. Donati and M. Norgia, “Self-mixing interferometry for biomedical signals sensing,” IEEE J. Sel. Top. Quantum Electron. 20(2), 104–111 (2014).

M. T. Fathi and S. Donati, “Thickness measurement of transparent plates by a self-mixing interferometer,” Opt. Lett. 35(11), 1844–1846 (2010).
[PubMed]

T. Bosch and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40(1), 20–27 (2001).

Fan, K. C.

Fan, Y.

Fathi, M. T.

Gao, W.

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

Guo, B.

Guo, D.

D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).

D. Guo and M. Wang, “Note: Design of a laser feedback interferometer with double diffraction system,” Rev. Sci. Instrum. 86(9), 096111 (2015).
[PubMed]

W. Xia, M. Wang, Z. Yang, W. Guo, H. Hao, and D. Guo, “High-accuracy sinusoidal phase-modulating self-mixing interferometer using an electro-optic modulator: development and evaluation,” Appl. Opt. 52(4), B52–B59 (2013).
[PubMed]

Guo, Q.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Guo, W.

Hao, H.

D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).

W. Xia, M. Wang, Z. Yang, W. Guo, H. Hao, and D. Guo, “High-accuracy sinusoidal phase-modulating self-mixing interferometer using an electro-optic modulator: development and evaluation,” Appl. Opt. 52(4), B52–B59 (2013).
[PubMed]

Hsu, C. C.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

Kimura, A.

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

Lee, J. Y.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

Li, H.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Lin, K.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Lu, Y.

Norgia, M.

S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).

S. Donati and M. Norgia, “Self-mixing interferometry for biomedical signals sensing,” IEEE J. Sel. Top. Quantum Electron. 20(2), 104–111 (2014).

Rossi, D.

S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).

Su, L.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Sun, L.

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

Tan, Y.

K. Zhu, B. Guo, Y. Lu, S. Zhang, and Y. Tan, “Single-spot two-dimensional displacement measurement based on self-mixing interferometry,” Optica 4(7), 729–735 (2017).

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009).
[PubMed]

Tong, J.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Wang, M.

D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).

D. Guo and M. Wang, “Note: Design of a laser feedback interferometer with double diffraction system,” Rev. Sci. Instrum. 86(9), 096111 (2015).
[PubMed]

W. Xia, M. Wang, Z. Yang, W. Guo, H. Hao, and D. Guo, “High-accuracy sinusoidal phase-modulating self-mixing interferometer using an electro-optic modulator: development and evaluation,” Appl. Opt. 52(4), B52–B59 (2013).
[PubMed]

Wang, W.

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

Weng, H. F.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

Wu, C. C.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

Xi, J.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Y. Fan, Y. Yu, J. Xi, and J. F. Chicharo, “Improving the measurement performance for a self-mixing interferometry-based displacement sensing system,” Appl. Opt. 50(26), 5064–5072 (2011).
[PubMed]

Xia, W.

Xu, C.

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

Yang, Z.

Yu, Y.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Y. Fan, Y. Yu, J. Xi, and J. F. Chicharo, “Improving the measurement performance for a self-mixing interferometry-based displacement sensing system,” Appl. Opt. 50(26), 5064–5072 (2011).
[PubMed]

Zeng, L.

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

Zhang, S.

K. Zhu, B. Guo, Y. Lu, S. Zhang, and Y. Tan, “Single-spot two-dimensional displacement measurement based on self-mixing interferometry,” Optica 4(7), 729–735 (2017).

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009).
[PubMed]

Zhang, Y.

Zhao, S.

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

Zhu, K.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Tan, S. Zhang, C. Xu, and S. Zhao, “Inspecting and locating foreign body in biological sample by laser confocal feedback technology,” Appl. Phys. Lett. 103(10), 101909 (2013).

IEEE J. Quantum Electron. (1)

S. Donati, D. Rossi, and M. Norgia, “Single channel self-mixing interferometer measures simultaneously displacement and tilt and yaw angles of a reflective target,” IEEE J. Quantum Electron. 51(12), 1–8 (2015).

IEEE J. Sel. Top. Quantum Electron. (1)

S. Donati and M. Norgia, “Self-mixing interferometry for biomedical signals sensing,” IEEE J. Sel. Top. Quantum Electron. 20(2), 104–111 (2014).

IEEE Photonics Technol. Lett. (1)

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Spectrum broadening in optical frequency-shifted feedback of microchip laser,” IEEE Photonics Technol. Lett. 28(14), 1593–1596 (2016).

Opt. Commun. (1)

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008).

Opt. Eng. (1)

T. Bosch and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40(1), 20–27 (2001).

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Precis. Eng. (1)

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).

Proc. SPIE (1)

D. Guo, M. Wang, and H. Hao, “Self-mixing grating interferometer: theoretical analysis and experimental observations,” Proc. SPIE 9960, 996019 (2016).

Rev. Sci. Instrum. (1)

D. Guo and M. Wang, “Note: Design of a laser feedback interferometer with double diffraction system,” Rev. Sci. Instrum. 86(9), 096111 (2015).
[PubMed]

Sci. Rep. (1)

Y. Tan, W. Wang, C. Xu, and S. Zhang, “Laser confocal feedback tomography and nano-step height measurement,” Sci. Rep. 3, 2971 (2013).
[PubMed]

Sensor Actuat. A-Phys. (1)

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sensor Actuat. A-Phys. 137(1), 185–191 (2007).

Sensors (Basel) (1)

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[PubMed]

Other (2)

C. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

S. Zhang and W. Holzapfel, Orthogonal Polarization in Lasers: Physical Phenomena and Engineering Applications (John Wiley and Sons, 2013).

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of the SMGI with auto-collimation design.
Fig. 2
Fig. 2 Simulation results of the SMGI signal (vibration amplitude at 4μm p-p).
Fig. 3
Fig. 3 Schematic diagram of the a SMGI with multiple-diffraction.
Fig. 4
Fig. 4 Schematic diagram of the beam path between the grating and the mirror.
Fig. 5
Fig. 5 Experimental Setup of a SMGI with multiple-diffraction.
Fig. 6
Fig. 6 Experimental observations of the SMGI signal with multiple-diffraction (N = 1, 3, 5).
Fig. 7
Fig. 7 Experimental observations of the SMGI signal with multiple-diffraction (N = 2, 4, 6).
Fig. 8
Fig. 8 Displacement measurement configuration using SMGI with multiple-diffraction.
Fig. 9
Fig. 9 Signal processing diagram of the phase extraction.
Fig. 10
Fig. 10 Reconstructed sinusoidal movements of the grating (10HZ).
Fig. 11
Fig. 11 Reconstructed triangular movements of the grating (10HZ).
Fig. 12
Fig. 12 Measurement result of the grating displacement with amplitude 5nm (p-p).

Tables (2)

Tables Icon

Table 1 Measurement results of the sinusoidal movements

Tables Icon

Table 2 Measurement results of the triangular movements

Equations (22)

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

θ = arc sin ( λ 2 d ) .
d ( sin θ + sin β q ) = q λ , q = ± 1 , ± 2... .
β 1 = θ .
Δ f = f 1 f 0 = 2 f 0 v c sin θ .
Δ f = v / d .
φ g ( t ) = 2 π Δ f t = 2 π x ( t ) / d .
ω τ = ω 0 τ C sin ( ω τ + 2 π x ( t ) d + a r c tan α ) .
I ( t ) = I 0 [ 1 + m cos ( ω τ + 2 π x ( t ) d ) ] .
φ g ( t )= 2 N π x ( t ) / d .
| Δ φ g ( t ) | = | 2 N π Δ x ( t ) / d | = 2 π | Δ x ( t ) | = d / N .
{ sin θ i + sin β i = λ / d , i = 1 , 2.. n θ i + 1 + β i = 2 α , i = 1 , 2.. n 1 θ n = β n = arc sin ( λ / 2 d ) . .
{ sin θ i + sin β i = λ / d , i = 1 , 2.. n θ i + 1 + β i = 2 α , i = 1 , 2.. n 1 β n = α . .
I ( t ) = I 0 { 1 + m cos [ ϕ 0 + φ g ( t ) + 2 a sin ( 2 π f m t ) ] } .
I ( f m , t ) = 2 m I 0 sin ( φ g ( t ) + ϕ 0 ) J 1 ( 2 a ) sin ( 2 π f m t ) = A 1 ( t ) sin ( 2 π f m t ) .
I ( 2 f m , t ) = 2 m I 0 cos ( φ g ( t ) + ϕ 0 ) J 2 ( 2 a ) cos ( 4 π f m t ) = A 2 ( t ) cos ( 4 π f m t ) .
Δ φ g ( t ) = Δ arc tan [ A 1 ( t ) A 2 ( t ) J 2 ( 2 a ) J 1 ( 2 a ) ] .
Δ x ( t ) = d 2 π N Δ φ g ( t ) .
s = d φ g d x = 2 N π d .
| Δ x | = d 2 N π | Δ φ g | + φ g 2 N π | Δ d | .
d φ g 2 π d t f m 2 .
v max = f m d / 2 N .
x 0 max = f m d / 4 π N f 0 .

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