Abstract

Phase measurement of interference fringes is an integral part of several fields in optics. Using simple straight sinusoidal fringe patterns, we describe the relationship between fringe position or phase to the centroid position when these fringes are incident on a position sensitive detector. With detailed descriptions and some experimental results, we show that a phenomenal sensitivity is possible in principle with what we believe is a new approach of phase measurement, and excellent sensitivity is readily achieved.

© 2007 Optical Society of America

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References

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  1. A. K. Poteomkin, A. N. Mal'shakov, and N. F. Andreev, "Use of self-focusing for measurements of ultrasmall (less than λ/3000) wave-front distortions," J. Opt. Soc. Am. B 19, 650-655 (2002).
    [CrossRef]
  2. C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L.-Z. Cai, and Y.-R. Wang, "Determination of global phase shifts between interferograms by use of an energy-minimum algorithm," Appl. Opt. 42, 6514-6519 (2003).
    [CrossRef] [PubMed]
  3. See, for example, H. J. Caulfield, "Centroid detection in metrology," Opt. Commun. 36, 439-440 (1981).
    [CrossRef]
  4. S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
    [CrossRef]
  5. C. S. Vikram and H. J. Caulfield, "Position-sensing detector for logical operations in incoherent light," Opt. Eng. 44, 115201-1-4 (2005).
  6. J. D. Valentine and A. E. Rana, "Centroid and full-width at half maximum uncertainties of histogrammed data with an underlying Gaussian distribution--the moments method," IEEE Trans. Nucl. Sci. 43, 2501-2508 (1996).
    [CrossRef]
  7. S. Hosoe, "Laser interferometric system for displacement measurement with high precision," Nanotechnology 2, 88-95 (1991).
    [CrossRef]
  8. S. Hosoe, "Highly precise and stable laser displacement measurement interferometer with differential optical passes in practical use," Nanotechnology 4, 81-85 (1993).
    [CrossRef]
  9. F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, "Absolute distance measurements by variable wavelength interferometry," Appl. Opt. 20, 400-403 (1981).
    [CrossRef] [PubMed]
  10. P. J. Bryanston-Cross and Z. Wang, "Camera focusing based on fringe pattern matching," Appl. Opt. 36, 6498-6502 (1997).
    [CrossRef]
  11. Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
    [CrossRef]
  12. R. S. Sirohi and M. P. Kothiyal, Optical Components, Systems, and Measurement Techniques (Marcel Dekker, 1991).
  13. Y.-Y. Cheng and J. C. Wyant, "Phase shifter calibration in phase-shifting interferometry," Appl. Opt. 24, 3049-3052 (1985).
    [CrossRef] [PubMed]
  14. H. van Brug, "Phase-step calibration for phase-stepped interferometry," Appl. Opt. 38, 3549-3555 (1999).
    [CrossRef]
  15. L.-Z. Cai, Q. Liu, Y.-R. Wang, X.-F. Meng, and M. Z. He, "Experimental demonstrations of the digital correlation of complex wave errors caused by arbitrary phase-shift errors in phase-shifting interferometry," Appl. Opt. 45, 1193-1202 (2006).
    [CrossRef] [PubMed]
  16. C. S. Vikram, "Holography of erosion, corrosion, and mechanical wear: possible role of phase-shifting interferometry," Opt. Eng. 35, 1795-1796 (1996).
    [CrossRef]
  17. H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
    [CrossRef]
  18. Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
    [CrossRef]
  19. N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
    [CrossRef]
  20. H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
    [CrossRef]
  21. C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
    [CrossRef]
  22. Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
    [CrossRef]
  23. G. Srikanth, "Measurement of period of interference patterns with submicron period," Opt. Laser Technol. 39, 918-921 (2007).
    [CrossRef]

2007

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

G. Srikanth, "Measurement of period of interference patterns with submicron period," Opt. Laser Technol. 39, 918-921 (2007).
[CrossRef]

2006

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

L.-Z. Cai, Q. Liu, Y.-R. Wang, X.-F. Meng, and M. Z. He, "Experimental demonstrations of the digital correlation of complex wave errors caused by arbitrary phase-shift errors in phase-shifting interferometry," Appl. Opt. 45, 1193-1202 (2006).
[CrossRef] [PubMed]

2005

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

2004

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

2003

2002

A. K. Poteomkin, A. N. Mal'shakov, and N. F. Andreev, "Use of self-focusing for measurements of ultrasmall (less than λ/3000) wave-front distortions," J. Opt. Soc. Am. B 19, 650-655 (2002).
[CrossRef]

Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
[CrossRef]

2001

H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
[CrossRef]

1999

1998

S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
[CrossRef]

1997

1996

J. D. Valentine and A. E. Rana, "Centroid and full-width at half maximum uncertainties of histogrammed data with an underlying Gaussian distribution--the moments method," IEEE Trans. Nucl. Sci. 43, 2501-2508 (1996).
[CrossRef]

C. S. Vikram, "Holography of erosion, corrosion, and mechanical wear: possible role of phase-shifting interferometry," Opt. Eng. 35, 1795-1796 (1996).
[CrossRef]

Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
[CrossRef]

1993

S. Hosoe, "Highly precise and stable laser displacement measurement interferometer with differential optical passes in practical use," Nanotechnology 4, 81-85 (1993).
[CrossRef]

1991

S. Hosoe, "Laser interferometric system for displacement measurement with high precision," Nanotechnology 2, 88-95 (1991).
[CrossRef]

R. S. Sirohi and M. P. Kothiyal, Optical Components, Systems, and Measurement Techniques (Marcel Dekker, 1991).

1985

1981

Alonso, J.

H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
[CrossRef]

Andreev, N. F.

Bachor, H.-A.

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

Bernabeu, E.

H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
[CrossRef]

Bien, F.

Bryanston-Cross, P. J.

P. J. Bryanston-Cross and Z. Wang, "Camera focusing based on fringe pattern matching," Appl. Opt. 36, 6498-6502 (1997).
[CrossRef]

Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
[CrossRef]

Cai, L.-Z.

Camac, M.

Canabal, H.

H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
[CrossRef]

Caulfield, H. J.

F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, "Absolute distance measurements by variable wavelength interferometry," Appl. Opt. 20, 400-403 (1981).
[CrossRef] [PubMed]

See, for example, H. J. Caulfield, "Centroid detection in metrology," Opt. Commun. 36, 439-440 (1981).
[CrossRef]

C. S. Vikram and H. J. Caulfield, "Position-sensing detector for logical operations in incoherent light," Opt. Eng. 44, 115201-1-4 (2005).

Chen, T. P.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Chen, Y. C.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Cheng, Y.-Y.

Chien, C. H.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Chiou, Y. T.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Ezekiel, S.

Fabre, C.

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

Fujigaki, M.

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

Gonzo, L.

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

Gottardi, M.

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

Guo, C.-S.

He, J.-L.

He, M. Z.

Hosoe, S.

S. Hosoe, "Highly precise and stable laser displacement measurement interferometer with differential optical passes in practical use," Nanotechnology 4, 81-85 (1993).
[CrossRef]

S. Hosoe, "Laser interferometric system for displacement measurement with high precision," Nanotechnology 2, 88-95 (1991).
[CrossRef]

Hsieh, C. C.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Kawagishi, N.

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

Kothiyal, M. P.

R. S. Sirohi and M. P. Kothiyal, Optical Components, Systems, and Measurement Techniques (Marcel Dekker, 1991).

Lam, P. K.

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

Lapone, A. M.

S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
[CrossRef]

Liu, Q.

Low, S. H.

S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
[CrossRef]

Mal'shakov, A. N.

Massari, N.

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

Matui, T.

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

Maxemchuk, N. F.

S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
[CrossRef]

Mendelovic, D.

Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
[CrossRef]

Meng, X.-F.

Morimoto, Y.

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

Poteomkin, A. K.

Rana, A. E.

J. D. Valentine and A. E. Rana, "Centroid and full-width at half maximum uncertainties of histogrammed data with an underlying Gaussian distribution--the moments method," IEEE Trans. Nucl. Sci. 43, 2501-2508 (1996).
[CrossRef]

Rong, Z.-Y.

Shrot, Y.

Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
[CrossRef]

Simoni, A.

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

Sirohi, R. S.

R. S. Sirohi and M. P. Kothiyal, Optical Components, Systems, and Measurement Techniques (Marcel Dekker, 1991).

Srikanth, G.

G. Srikanth, "Measurement of period of interference patterns with submicron period," Opt. Laser Technol. 39, 918-921 (2007).
[CrossRef]

Treps, N.

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

Tsai, M. L.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Valentine, J. D.

J. D. Valentine and A. E. Rana, "Centroid and full-width at half maximum uncertainties of histogrammed data with an underlying Gaussian distribution--the moments method," IEEE Trans. Nucl. Sci. 43, 2501-2508 (1996).
[CrossRef]

van Brug, H.

Vikram, C. S.

C. S. Vikram, "Holography of erosion, corrosion, and mechanical wear: possible role of phase-shifting interferometry," Opt. Eng. 35, 1795-1796 (1996).
[CrossRef]

C. S. Vikram and H. J. Caulfield, "Position-sensing detector for logical operations in incoherent light," Opt. Eng. 44, 115201-1-4 (2005).

Wang, C. T.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Wang, H.-T.

Wang, Y.-R.

Wang, Z.

P. J. Bryanston-Cross and Z. Wang, "Camera focusing based on fringe pattern matching," Appl. Opt. 36, 6498-6502 (1997).
[CrossRef]

Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
[CrossRef]

Whitehouse, D. J.

Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
[CrossRef]

Wu, Y. D.

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Wyant, J. C.

Zalevsky, Z.

Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
[CrossRef]

Appl. Opt.

Contemp. Phys.

H.-A. Bachor, C. Fabre, P. K. Lam, and N. Treps, "Teaching a laser beam to go straight," Contemp. Phys. 46, 395-405 (2005).
[CrossRef]

IEEE Trans. Commun.

S. H. Low, N. F. Maxemchuk, and A. M. Lapone, "Document identification for copyright protection using centroid detection," IEEE Trans. Commun. 46, 372-383 (1998).
[CrossRef]

IEEE Trans. Instrum. Meas.

N. Massari, L. Gonzo, M. Gottardi, and A. Simoni, "A fast CMOS optical position sensor with high subpixel resolution," IEEE Trans. Instrum. Meas. 53, 116-123 (2004).
[CrossRef]

IEEE Trans. Nucl. Sci.

J. D. Valentine and A. E. Rana, "Centroid and full-width at half maximum uncertainties of histogrammed data with an underlying Gaussian distribution--the moments method," IEEE Trans. Nucl. Sci. 43, 2501-2508 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Nanotechnology

S. Hosoe, "Laser interferometric system for displacement measurement with high precision," Nanotechnology 2, 88-95 (1991).
[CrossRef]

S. Hosoe, "Highly precise and stable laser displacement measurement interferometer with differential optical passes in practical use," Nanotechnology 4, 81-85 (1993).
[CrossRef]

Opt. Commun.

See, for example, H. J. Caulfield, "Centroid detection in metrology," Opt. Commun. 36, 439-440 (1981).
[CrossRef]

Opt. Eng.

C. S. Vikram and H. J. Caulfield, "Position-sensing detector for logical operations in incoherent light," Opt. Eng. 44, 115201-1-4 (2005).

C. S. Vikram, "Holography of erosion, corrosion, and mechanical wear: possible role of phase-shifting interferometry," Opt. Eng. 35, 1795-1796 (1996).
[CrossRef]

H. Canabal, J. Alonso, and E. Bernabeu, "Laser beam deflectometry based on a subpixel resolution algorithm," Opt. Eng. 40, 2517-2523 (2001).
[CrossRef]

Z. Zalevsky, Y. Shrot, and D. Mendelovic, "Novel denoising algorithm for obtaining a superresolved position estimation," Opt. Eng. 41, 1355-1357 (2002).
[CrossRef]

Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, "Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry," Opt. Eng. 46, 025603-1-8 (2007).
[CrossRef]

Opt. Laser Technol.

G. Srikanth, "Measurement of period of interference patterns with submicron period," Opt. Laser Technol. 39, 918-921 (2007).
[CrossRef]

Z. Wang, P. J. Bryanston-Cross, and D. J. Whitehouse, "Phase difference determination by fringe pattern matching," Opt. Laser Technol. 28, 417-422 (1996).
[CrossRef]

Optics and Lasers in Engineering

C. H. Chien, Y. D. Wu, Y. T. Chiou, C. C. Hsieh, Y. C. Chen, T. P. Chen, M. L. Tsai, and C. T. Wang, "Nanoscale deformation measurement by using the hybrid method of gray-level and holographic interferometry," Optics and Lasers in Engineering 44, 80-91 (2006).
[CrossRef]

Other

R. S. Sirohi and M. P. Kothiyal, Optical Components, Systems, and Measurement Techniques (Marcel Dekker, 1991).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram representing the coordinates system and length of the square position sensing detector.

Fig. 2
Fig. 2

Variation of the centroid position against δ for some values of N.

Fig. 3
Fig. 3

Variation of the centroid position against the fringe number N for some values of δ.

Fig. 4
Fig. 4

Sensitivity against phase for some fringe numbers.

Fig. 5
Fig. 5

Contrast effects on centroid position for N = 0.1 case.

Fig. 6
Fig. 6

Diagram representing the experimental setup.

Fig. 7
Fig. 7

The experimental setup.

Fig. 8
Fig. 8

Position versus phase plot. Details are described in the text.

Fig. 9
Fig. 9

Variations between experimental and theoretical values of the plot of Fig. 8.

Equations (43)

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

I ( x ) = 1 + γ cos ( 2 N π x Δ + δ ) ,
X ¯ = Δ / 2 Δ / 2 x I ( x ) d x Δ / 2 Δ / 2 I ( x ) d x .
X ¯ = γ Δ sin δ [ cos ( π N ) π N sin ( π N ) ( π N ) 2 ] 2 [ 1 + γ cos δ sin ( π N ) π N ] .
( γ = 1 )
Δ = 1 c m
N = 0.5
2626.58   μm
0 .50   μm
δ = π
δ = 0
N = 20
Δ = 1   cm
γ = 1
δ = π
0.002 π
0.50   μm
( d X ¯ / d δ )
δ = π
δ = π
1   cm × 1   cm
0 .50   μm
0 .25   μm
δ = π
7.62   cm
10   mW
632 .8   nm
50   mm
1   cm
2 ( 1 R ) / [ 1 + ( 1 R ) 2 ]
δ = 0
1   cm
100   μm
0.92
sin ( δ ) / ( 2 π )
46.98   μW
1 .96   μW
1500   μm
3   μm
0.18   μm
1   mW
633   nm
0 .1   nm
N = 0.1

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