Abstract

We present an innovative structure of a linear diffraction grating interferometer as a long stroke and nanometer resolution displacement sensor for any linear stage. The principle of this diffractive interferometer is based on the phase information encoded by the ±1st order beams diffracted by a holographic grating. Properly interfering these two beams leads to modulation similar to a Doppler frequency shift that can be translated to displacement measurements via phase decoding. A self-compensation structure is developed to improve the alignment tolerance. LightTool analysis shows that this new structure is completely immune to alignment errors of offset, standoff, yaw, and roll. The tolerance of the pitch is also acceptable for most installation conditions. In order to compact the structure and improve the signal quality, a new optical bonding technology by mechanical fixture is presented so that the miniature optics can be permanently bonded together without an air gap in between. For the output waveform signals, a software module is developed for fast real-time pulse counting and phase subdivision. A laser interferometer HP5529A is employed to test the repeatability of the whole system. Experimental data show that within 15mm travel length, the repeatability is within 15nm.

© 2011 Optical Society of America

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References

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  1. X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).
  2. J. A. Cramar, “Nanometer resolution metrology with the molecular measuring machine,” Meas. Sci. Technol. 16, 2121–2128 (2005).
    [CrossRef]
  3. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  4. T. Pozar, P. Gregorcic, and J. Mozina, “Optimization of displacement-measuring quadrature interferometers considering the real properties of optical components,” Appl. Opt. 50, 1210–1219 (2011).
    [CrossRef] [PubMed]
  5. F. M. Gerasimov, “Use of diffraction gratings for controlling of a ruling engine,” Appl. Opt. 6, 1861–1865 (1967).
    [CrossRef] [PubMed]
  6. M. L. Schattenburg and H. I. Smith, “The critical role of metrology in nanotechnology,” Proc. SPIE 4608, 116–124 (2002).
    [CrossRef]
  7. C. L. Chu and K. C. Fan, “Design of a digital controller for long-stroke submicron positioning stage,” in Proceedings of the 1st International Conference of Positioning Technology (ICPT, 2004), pp. 176–181.
  8. S. G. Rautian, “On the theory of interferometers with diffraction gratings,” Opt. Spectrosc. 93, 934–940 (2002).
    [CrossRef]
  9. A. Kozłowska, M. Kujawińska, and C. Gorecki, “Grating interferometry with a semiconductor light source,” Appl. Opt. 36, 8116–8120 (1997).
    [CrossRef]
  10. X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
    [CrossRef]
  11. K. C. Fan and C. D. Su, “Error analysis for a diffraction grating interferometric stylus probing system,” Meas. Sci. Technol. 12, 482–490 (2001).
    [CrossRef]
  12. K. C. Fan and Y. S. Liu, “A linear diffraction grating interferometer with high accuracy,” Proc. SPIE 6280, 628008(2006).
    [CrossRef]
  13. Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
    [CrossRef]
  14. K. W. Wang and L. J. Zeng, “Double-grating frequency shifter for low-coherence heterodyne interferometry,” Opt. Commun. 251, 1–5 (2005).
    [CrossRef]
  15. K. C. Fan and Z. F. Lai, “A displacement spindle in a micro/nano level,” Meas. Sci. Technol. 18, 1710–1717 (2007).
    [CrossRef]
  16. C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
    [CrossRef]
  17. P. Wen and D. H. Hsu, “Direct subdivision of moiré fringe with CCD,” Proc. SPIE 1230, 165–166 (1990).
  18. J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).
  19. K. C. Fan and Y. J. Chen, “A study on digital subdivision of linear optical encoder for nanopositioning,” in Proceedings of the 6th International Conference on Frontiers of Design and Manufacturing (Science Press, 2004).
  20. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
    [CrossRef]
  21. K. C. Fan and F. Cheng, “Nanopositioning control on a commercial linear stage by software error correction,” Nanotech. Precis. Eng. 4, 1–9 (2006).
  22. R. Guenther, Modern Optics (Wiley, 1990).
  23. K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
    [CrossRef]
  24. F. Cheng, “Study on the key technology of nano-CMM measurement and control system,” Ph.D. thesis (Hefei University of Technology, 2010).
  25. F. L. Petrotti and L. S. Petrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).
  26. E. Hecht, Optics (Addison-Wesley, 1998).

2011 (1)

2010 (1)

F. Cheng, “Study on the key technology of nano-CMM measurement and control system,” Ph.D. thesis (Hefei University of Technology, 2010).

2009 (1)

K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
[CrossRef]

2008 (2)

X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
[CrossRef]

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

2007 (1)

K. C. Fan and Z. F. Lai, “A displacement spindle in a micro/nano level,” Meas. Sci. Technol. 18, 1710–1717 (2007).
[CrossRef]

2006 (2)

K. C. Fan and Y. S. Liu, “A linear diffraction grating interferometer with high accuracy,” Proc. SPIE 6280, 628008(2006).
[CrossRef]

K. C. Fan and F. Cheng, “Nanopositioning control on a commercial linear stage by software error correction,” Nanotech. Precis. Eng. 4, 1–9 (2006).

2005 (2)

K. W. Wang and L. J. Zeng, “Double-grating frequency shifter for low-coherence heterodyne interferometry,” Opt. Commun. 251, 1–5 (2005).
[CrossRef]

J. A. Cramar, “Nanometer resolution metrology with the molecular measuring machine,” Meas. Sci. Technol. 16, 2121–2128 (2005).
[CrossRef]

2004 (2)

C. L. Chu and K. C. Fan, “Design of a digital controller for long-stroke submicron positioning stage,” in Proceedings of the 1st International Conference of Positioning Technology (ICPT, 2004), pp. 176–181.

K. C. Fan and Y. J. Chen, “A study on digital subdivision of linear optical encoder for nanopositioning,” in Proceedings of the 6th International Conference on Frontiers of Design and Manufacturing (Science Press, 2004).

2002 (3)

Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
[CrossRef]

S. G. Rautian, “On the theory of interferometers with diffraction gratings,” Opt. Spectrosc. 93, 934–940 (2002).
[CrossRef]

M. L. Schattenburg and H. I. Smith, “The critical role of metrology in nanotechnology,” Proc. SPIE 4608, 116–124 (2002).
[CrossRef]

2001 (1)

K. C. Fan and C. D. Su, “Error analysis for a diffraction grating interferometric stylus probing system,” Meas. Sci. Technol. 12, 482–490 (2001).
[CrossRef]

2000 (1)

J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).

1999 (1)

X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).

1998 (1)

E. Hecht, Optics (Addison-Wesley, 1998).

1997 (1)

1996 (1)

F. L. Petrotti and L. S. Petrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1990 (3)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

R. Guenther, Modern Optics (Wiley, 1990).

P. Wen and D. H. Hsu, “Direct subdivision of moiré fringe with CCD,” Proc. SPIE 1230, 165–166 (1990).

1967 (1)

Birch, K. P.

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Chen, Q.

J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).

Chen, X. H.

X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).

Chen, Y. J.

K. C. Fan and Y. J. Chen, “A study on digital subdivision of linear optical encoder for nanopositioning,” in Proceedings of the 6th International Conference on Frontiers of Design and Manufacturing (Science Press, 2004).

Cheng, F.

F. Cheng, “Study on the key technology of nano-CMM measurement and control system,” Ph.D. thesis (Hefei University of Technology, 2010).

K. C. Fan and F. Cheng, “Nanopositioning control on a commercial linear stage by software error correction,” Nanotech. Precis. Eng. 4, 1–9 (2006).

Chu, C. L.

C. L. Chu and K. C. Fan, “Design of a digital controller for long-stroke submicron positioning stage,” in Proceedings of the 1st International Conference of Positioning Technology (ICPT, 2004), pp. 176–181.

Chu, X. C.

X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
[CrossRef]

Cramar, J. A.

J. A. Cramar, “Nanometer resolution metrology with the molecular measuring machine,” Meas. Sci. Technol. 16, 2121–2128 (2005).
[CrossRef]

Cui, J.

J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).

Fan, K. C.

K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
[CrossRef]

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

K. C. Fan and Z. F. Lai, “A displacement spindle in a micro/nano level,” Meas. Sci. Technol. 18, 1710–1717 (2007).
[CrossRef]

K. C. Fan and Y. S. Liu, “A linear diffraction grating interferometer with high accuracy,” Proc. SPIE 6280, 628008(2006).
[CrossRef]

K. C. Fan and F. Cheng, “Nanopositioning control on a commercial linear stage by software error correction,” Nanotech. Precis. Eng. 4, 1–9 (2006).

K. C. Fan and Y. J. Chen, “A study on digital subdivision of linear optical encoder for nanopositioning,” in Proceedings of the 6th International Conference on Frontiers of Design and Manufacturing (Science Press, 2004).

C. L. Chu and K. C. Fan, “Design of a digital controller for long-stroke submicron positioning stage,” in Proceedings of the 1st International Conference of Positioning Technology (ICPT, 2004), pp. 176–181.

K. C. Fan and C. D. Su, “Error analysis for a diffraction grating interferometric stylus probing system,” Meas. Sci. Technol. 12, 482–490 (2001).
[CrossRef]

Gerasimov, F. M.

Gorecki, C.

Gregorcic, P.

Guenther, R.

R. Guenther, Modern Optics (Wiley, 1990).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

Hsu, D. H.

P. Wen and D. H. Hsu, “Direct subdivision of moiré fringe with CCD,” Proc. SPIE 1230, 165–166 (1990).

Jay, J.

Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
[CrossRef]

Jourlin, Y.

Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
[CrossRef]

Kao, C. F.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

Kozlowska, A.

Kujawinska, M.

Lai, Z. F.

K. C. Fan and Z. F. Lai, “A displacement spindle in a micro/nano level,” Meas. Sci. Technol. 18, 1710–1717 (2007).
[CrossRef]

Li, B. K.

K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
[CrossRef]

Li, D. C.

X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).

Li, H. Q.

J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).

Liu, C. H.

K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
[CrossRef]

Liu, Y. S.

K. C. Fan and Y. S. Liu, “A linear diffraction grating interferometer with high accuracy,” Proc. SPIE 6280, 628008(2006).
[CrossRef]

Lu, S. H.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

Lü, H. B.

X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
[CrossRef]

Mozina, J.

Parriaux, O.

Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
[CrossRef]

Petrotti, F. L.

F. L. Petrotti and L. S. Petrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

Petrotti, L. S.

F. L. Petrotti and L. S. Petrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

Pozar, T.

Rautian, S. G.

S. G. Rautian, “On the theory of interferometers with diffraction gratings,” Opt. Spectrosc. 93, 934–940 (2002).
[CrossRef]

Schattenburg, M. L.

M. L. Schattenburg and H. I. Smith, “The critical role of metrology in nanotechnology,” Proc. SPIE 4608, 116–124 (2002).
[CrossRef]

Shen, H. M.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

Smith, H. I.

M. L. Schattenburg and H. I. Smith, “The critical role of metrology in nanotechnology,” Proc. SPIE 4608, 116–124 (2002).
[CrossRef]

Su, C. D.

K. C. Fan and C. D. Su, “Error analysis for a diffraction grating interferometric stylus probing system,” Meas. Sci. Technol. 12, 482–490 (2001).
[CrossRef]

Wang, K. W.

K. W. Wang and L. J. Zeng, “Double-grating frequency shifter for low-coherence heterodyne interferometry,” Opt. Commun. 251, 1–5 (2005).
[CrossRef]

Wen, P.

P. Wen and D. H. Hsu, “Direct subdivision of moiré fringe with CCD,” Proc. SPIE 1230, 165–166 (1990).

Zeng, L. J.

K. W. Wang and L. J. Zeng, “Double-grating frequency shifter for low-coherence heterodyne interferometry,” Opt. Commun. 251, 1–5 (2005).
[CrossRef]

Zhao, S. H.

X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
[CrossRef]

Zhao, Y.

X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).

Appl. Opt. (3)

Jpn. J. Appl. Phys. (1)

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47, 1833–1837 (2008).
[CrossRef]

Meas. Sci. Technol. (5)

X. C. Chu, H. B. Lü, and S. H. Zhao, “Research on long-range grating interferometry with nanometer resolution,” Meas. Sci. Technol. 19, 017001 (2008).
[CrossRef]

K. C. Fan and C. D. Su, “Error analysis for a diffraction grating interferometric stylus probing system,” Meas. Sci. Technol. 12, 482–490 (2001).
[CrossRef]

K. C. Fan and Z. F. Lai, “A displacement spindle in a micro/nano level,” Meas. Sci. Technol. 18, 1710–1717 (2007).
[CrossRef]

J. A. Cramar, “Nanometer resolution metrology with the molecular measuring machine,” Meas. Sci. Technol. 16, 2121–2128 (2005).
[CrossRef]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Nanotech. Precis. Eng. (1)

K. C. Fan and F. Cheng, “Nanopositioning control on a commercial linear stage by software error correction,” Nanotech. Precis. Eng. 4, 1–9 (2006).

Opt. Commun. (1)

K. W. Wang and L. J. Zeng, “Double-grating frequency shifter for low-coherence heterodyne interferometry,” Opt. Commun. 251, 1–5 (2005).
[CrossRef]

Opt. Spectrosc. (1)

S. G. Rautian, “On the theory of interferometers with diffraction gratings,” Opt. Spectrosc. 93, 934–940 (2002).
[CrossRef]

Opt. Tech. (2)

X. H. Chen, Y. Zhao, and D. C. Li, “Review on optical nanometrology,” Opt. Tech. 3, 74–78 (1999).

J. Cui, H. Q. Li, and Q. Chen, “New digital subdividing and raster-sensing technique for moiré fringe and gratings,” Opt. Tech. 26, 294–296 (2000).

Precis. Eng. (2)

Y. Jourlin, J. Jay, and O. Parriaux, “Compact diffractive interferometric displacement sensor in reflection,” Precis. Eng. 26, 1–6 (2002).
[CrossRef]

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

Proc. SPIE (4)

K. C. Fan, B. K. Li, and C. H. Liu, “A diffraction grating scale for long range and nanometer resolution,” Proc. SPIE 7133, J1–J8 (2009).
[CrossRef]

K. C. Fan and Y. S. Liu, “A linear diffraction grating interferometer with high accuracy,” Proc. SPIE 6280, 628008(2006).
[CrossRef]

P. Wen and D. H. Hsu, “Direct subdivision of moiré fringe with CCD,” Proc. SPIE 1230, 165–166 (1990).

M. L. Schattenburg and H. I. Smith, “The critical role of metrology in nanotechnology,” Proc. SPIE 4608, 116–124 (2002).
[CrossRef]

Other (6)

C. L. Chu and K. C. Fan, “Design of a digital controller for long-stroke submicron positioning stage,” in Proceedings of the 1st International Conference of Positioning Technology (ICPT, 2004), pp. 176–181.

K. C. Fan and Y. J. Chen, “A study on digital subdivision of linear optical encoder for nanopositioning,” in Proceedings of the 6th International Conference on Frontiers of Design and Manufacturing (Science Press, 2004).

F. Cheng, “Study on the key technology of nano-CMM measurement and control system,” Ph.D. thesis (Hefei University of Technology, 2010).

F. L. Petrotti and L. S. Petrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

E. Hecht, Optics (Addison-Wesley, 1998).

R. Guenther, Modern Optics (Wiley, 1990).

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Figures (9)

Fig. 1
Fig. 1

Improved design of LDGI (LD, laser diode; G, grating; SH, shield; PBSi, ith polarizing beam splitter; Mi, ith mirror; NPBS, nonpolarizing beam splitter; Qi, ith quarter-wave plate; PDi, ith photodetector).

Fig. 2
Fig. 2

Typical geometric errors.

Fig. 3
Fig. 3

Interfering spots.

Fig. 4
Fig. 4

Harmonic disturbance caused by redundant reflections.

Fig. 5
Fig. 5

Compact LDGI system bonded with a mechanical fixture. (a) Optics in clamping; (b) actual dimensions (in mm).

Fig. 6
Fig. 6

Typical signal distortions.

Fig. 7
Fig. 7

Lissajous signals before (left) and after (right) vector operations.

Fig. 8
Fig. 8

Pulse counting in forward (left) and reverse (right) motions.

Fig. 9
Fig. 9

Noise-immune principle.

Tables (4)

Tables Icon

Table 1 Mathematical Expression of Some Typical Polarizations

Tables Icon

Table 2 Spots Moving Situation and Tolerance

Tables Icon

Table 3 Experimental Data of Displacement Measurements

Tables Icon

Table 4 Experimental Data after the Third-Order Polynomial Correction

Equations (23)

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

M G = [ k P ( θ ) 0 0 k S ( θ ) ] .
M Q 1 M 1 G M 1 G 1 = 2 2 [ 1 i i 1 ] [ k P ( θ ) 0 0 k S ( θ ) ] 2 2 [ 1 i i 1 ] = 1 2 [ k P ( θ ) k S ( θ ) i ( k P ( θ ) k S ( θ ) ) i ( k P ( θ ) k S ( θ ) ) k P ( θ ) + k S ( θ ) ] = 1 2 [ a i b i b a ] ,
{ a = k P ( θ ) + k S ( θ ) b = k P ( θ ) k S ( θ ) .
E L = 2 2 [ 1 i i 1 ] [ 1 0 0 0 ] 1 2 [ a i b i b a ] [ 0 0 0 1 ] [ 1 1 ] = 2 b i 4 [ 1 i ] ,
E R = 2 2 [ 1 i i 1 ] [ 0 0 0 1 ] 1 2 [ a i b i b a ] [ 1 0 0 0 ] [ 1 1 ] = 2 b 4 [ 1 i ] .
{ Δ f = f 0 v c ( sin θ i + sin θ q ) d ( sin θ i + sin θ q ) = m λ c = λ f 0 .
Δ f = m v d = ± v d ,
Δ ω = 2 π Δ f = ± 2 π v d .
E L = A · exp [ i ( ω Δ ω ) t ] [ 1 i ] ,
E R = A · exp [ i ( ω + Δ ω ) t ] [ 1 i ] .
E P D 1 = [ 0 0 0 1 ] ( E L + E R ) = [ 0 1 ] 2 A exp ( i ω t ) sin ( Δ ω · t ) ,
E P D 2 = [ 1 0 0 0 ] ( E L + E R ) = [ 1 0 ] 2 A exp ( i ω t ) cos ( Δ ω · t ) ,
E P D 3 = 1 2 [ 1 1 1 1 ] ( E L + E R ) = [ 1 1 ] A exp ( i ω t ) [ cos ( Δ ω · t ) + sin ( Δ ω · t ) ] ,
E P D 4 = 1 2 [ 1 1 1 1 ] ( E L + E R ) = [ 1 1 ] A exp ( i ω t ) [ cos ( Δ ω · t ) sin ( Δ ω · t ) ] .
I P D 1 = E P D 1 = A [ 1 cos ( 2 Δ ω · t ) ] = A [ 1 cos ( 2 π · 2 v d · t ) ] = A [ 1 cos ( 2 π · 2 s d ) ] ,
I P D 2 = E P D 2 = A [ 1 + cos ( 2 Δ ω · t ) ] = A [ 1 + cos ( 2 π · 2 v d · t ) ] = A [ 1 + cos ( 2 π · 2 s d ) ] ,
I P D 3 = E P D 3 = A [ 1 + sin ( 2 Δ ω · t ) ] = A [ 1 + sin ( 2 π · 2 v d · t ) ] = A [ 1 + sin ( 2 π · 2 s d ) ] ,
I P D 4 = E P D 4 = A [ 1 sin ( 2 Δ ω · t ) ] = A [ 1 sin ( 2 π · 2 v d · t ) ] = A [ 1 sin ( 2 π · 2 s d ) ] .
C = S 1 S 2 = π r 2 · π 2 θ 2 π × 2 r 2 sin 2 θ π r 2 × 100 % ,
θ = arcsin d 2 r .
u ( x ) = u ( x ) + v ( x ) = sin ( x ) + sin ( x + φ ) = 2 sin 2 x + φ 2 cos φ 2 ,
v ( x ) = u ( x ) v ( x ) = sin ( x ) sin ( x + φ ) = 2 cos 2 x + φ 2 sin φ 2 .
d c = 0.19 x 3 3.59 x 2 + 100.22 x 6.04.

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