Abstract

We present a novel microscope interferometric technique based on the heterodinization of two Gaussian beams for measuring roughness of optical surfaces in microscopic areas. One of the beams is used as a probe beam, focussed and reflected by the surface under test. The second beam interferes with the first beam and introduces a time varying modulating signal. The modulating light beam is obtained from the first diffraction order of a Bragg cell. The two beams are superimposed and added coherently at the sensitive plane of a photodetector that integrates the overall intensity of the beams. We show analytically that it is possible to find appropriate working conditions in which the system has a linear response. Under these conditions, the size of the probe beam at the plane of detection as well as the amplitude of the time varying signal at the output of the photodetector, are both proportional to the local vertical height of the surface under test. As a narrow bandwidth amplifier is used to detect the time varying signal the system exhibits a high signal to noise ratio. We also include experimental results of the measurement of the topography of a sample consisting in a blazed-reflecting grating.

© 2007 Optical Society of America

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  1. J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. 24, 380–387 (1985).
  2. J. M. Bennett and J. H. Dancy, “Stylus profiling instrument for measuring statistical properties of smooth optical surfaces,” Appl. Opt. 20, 1785 (1981).
    [CrossRef] [PubMed]
  3. D. Walker, H. Yang, and S. Kim, “Novel hybrid stylus for nanometric profilometry for large optical surfaces,” Opt. Express 11, 1793–1798 (2003).
    [CrossRef] [PubMed]
  4. H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
    [CrossRef]
  5. S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. 40, 2845 (2001).
    [CrossRef]
  6. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  7. W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
    [CrossRef]
  8. M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
    [CrossRef]
  9. Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
    [CrossRef]
  10. G. E. Sommargren, “Optical heterodyne profilometry,” Appl. Opt. 20, 335–343 (1981).
    [CrossRef]
  11. C-C. Huang, “Optical heterodyne profilometer,” Opt. Eng. 23, 365–370 (1984).
  12. J. C. Wyant, “Optical profilers for surface roughness,” Proc. SPIE 525, 174–180 (1985).
  13. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).
  14. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [CrossRef] [PubMed]
  15. G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).
  16. K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. 29, 3823–3827 (1990).
    [CrossRef] [PubMed]
  17. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [CrossRef] [PubMed]
  18. B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
    [CrossRef]
  19. M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
    [CrossRef]
  20. J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
    [CrossRef]
  21. M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).
  22. W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.
  23. A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
    [CrossRef]

2005 (1)

M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
[CrossRef]

2003 (3)

B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
[CrossRef]

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
[CrossRef]

D. Walker, H. Yang, and S. Kim, “Novel hybrid stylus for nanometric profilometry for large optical surfaces,” Opt. Express 11, 1793–1798 (2003).
[CrossRef] [PubMed]

2001 (2)

S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. 40, 2845 (2001).
[CrossRef]

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

2000 (2)

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).

1997 (1)

W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
[CrossRef]

1995 (1)

M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
[CrossRef]

1993 (1)

1990 (3)

1989 (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

1987 (1)

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

1985 (2)

J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. 24, 380–387 (1985).

J. C. Wyant, “Optical profilers for surface roughness,” Proc. SPIE 525, 174–180 (1985).

1984 (1)

C-C. Huang, “Optical heterodyne profilometer,” Opt. Eng. 23, 365–370 (1984).

1981 (2)

1977 (1)

G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).

Aguilar, J. F.

M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
[CrossRef]

Barrientos, B.

M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
[CrossRef]

B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
[CrossRef]

Bennett, J. M.

J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. 24, 380–387 (1985).

J. M. Bennett and J. H. Dancy, “Stylus profiling instrument for measuring statistical properties of smooth optical surfaces,” Appl. Opt. 20, 1785 (1981).
[CrossRef] [PubMed]

Caber, P. J.

Chi, G.

W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
[CrossRef]

Chim, S. S. C.

Clark, S. R.

S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. 40, 2845 (2001).
[CrossRef]

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

Creath, K.

Cywiak, M.

M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
[CrossRef]

B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
[CrossRef]

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).

Dancy, J. H.

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

Garnaes, J.

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
[CrossRef]

Goodman, W. J.

W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.

Greivenkap, J. E.

S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. 40, 2845 (2001).
[CrossRef]

Huang, C-C.

C-C. Huang, “Optical heterodyne profilometer,” Opt. Eng. 23, 365–370 (1984).

Johnson, G. W.

G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

Kim, S.

Kino, G. S.

Kühle, A.

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
[CrossRef]

Lee, B. S.

Leiner, D. C.

G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).

Li, W.

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

Moore, D. T.

G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).

Murakowski, J.

M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

Rosén, B.

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
[CrossRef]

Rosner, B.

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

See, C. W.

M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
[CrossRef]

Servín, M.

B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
[CrossRef]

Somekh, M. G.

M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
[CrossRef]

Sommargren, G. E.

Strand, T. C.

Suddendorf, M. B.

M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

van der Weide, D.

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

Wade, G.

M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).

Walker, D.

Wyant, J. C.

K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. 29, 3823–3827 (1990).
[CrossRef] [PubMed]

J. C. Wyant, “Optical profilers for surface roughness,” Proc. SPIE 525, 174–180 (1985).

Yang, H.

Zhang, T.

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

Zhou, W.

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
[CrossRef]

Zhou, Z.

W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
[CrossRef]

Zhou, Z. F.

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett (1)

M. B. Suddendorf, C. W. See, and M. G. Somekh, “Combined differential amplitude and phase interferometer with a single probe beam,” Appl. Phys. Lett,  67, 28–30 (1995).
[CrossRef]

IJIST (1)

M. Cywiak, J. Murakowski, and G. Wade., “Beam blocking method for optical characterization of surfaces,” IJIST 11, 164–169 (2000).

Opt. Commun. (1)

J. Murakowski, M. Cywiak, B. Rosner, and D. van der Weide, “Far field optical imaging with subwavelength resolution,” Opt. Commun. 185, 295–303 (2000).
[CrossRef]

Opt. Eng. (7)

Z. F. Zhou, T. Zhang, W. Zhou, and W. Li “Profilometer for measuring superfine surfaces,” Opt. Eng. 40, 1646–1652 (2001).
[CrossRef]

W. Zhou, Z. Zhou, and G. Chi, “Investigation of common-path interference profilometry,” Opt. Eng. 36, 3172–3175 (1997).
[CrossRef]

J. M. Bennett, “Comparison of techniques for measuring the roughness of optical surfaces,” Opt. Eng. 24, 380–387 (1985).

S. R. Clark and J. E. Greivenkap, “Optical reference profilometry,” Opt. Eng. 40, 2845 (2001).
[CrossRef]

B. Barrientos, M. Cywiak, and M. Servín, “Profilometry of optically smooth surfaces by a Gaussian probe beam,” Opt. Eng. 42, 3004–3012 (2003).
[CrossRef]

M. Cywiak, J. F. Aguilar, and B. Barrientos, “Low-numerical-aperture Gaussian beam confocal system for profiling optically smooth,” Opt. Eng. 44, 1–7 (2005).
[CrossRef]

C-C. Huang, “Optical heterodyne profilometer,” Opt. Eng. 23, 365–370 (1984).

Opt. Express (1)

Opt. Quantum Electron. (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

Proc. SPIE (4)

J. C. Wyant, “Optical profilers for surface roughness,” Proc. SPIE 525, 174–180 (1985).

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233–247 (1987).

A. Kühle, B. Rosén, and J. Garnaes, “Comparison of roughness measurement with atomic force microscopy and interference microscopy,” Proc. SPIE 5188, 154–161 (2003).
[CrossRef]

G. W. Johnson, D. C. Leiner, and D. T. Moore, “Phase-locked Interferometry,” Proc. SPIE 126, 152–160 (1977).

Other (1)

W. J. Goodman, Introduction to Fourier Optics, Second ed., Mc Graw-Hill, New York, 2000. Chap. 4, 5.

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Figures (6)

Fig. 1.
Fig. 1.

Experimental setup. (x 0, y 0) are the coordinates of a plane at the output of the Bragg-cell. (x, y) are the coordinates of the focal plane I of lens L1. (x 1, y 1) are the coordinates of the focal plane II. (x 2, y 2) are the coordinates of the focal plane II when the beam reaches again this plane after being reflected from the object under test. (x F, y F) are the coordinates of the focal plane I when the beam reaches this plane again in its way towards the photodedector. Finally, (ξ, η) are the coordinates at the plane of detection. M1 and M2 are mirrors, BS1 and BS2 are 50-50 beam splitters, and L1 is the focusing lens. For the description, the distance between the focal plane II and the object plane has been exaggerated.

Fig. 2.
Fig. 2.

Absolute amplitude distribution of the Gaussian beams at the plane of detection.

Fig. 3.
Fig. 3.

Semi-widths of the probe (continuos trace) and modulating (dotted line) beams at the detection plane, (ξ, η), as functions of the defocusing distance zp .

Fig. 4.
Fig. 4.

Total collected power as a function of the defocusing distance zp . The operating point is selected around the value zp = 3μm, within the range marked by the little segments on the graph.

Fig. 5.
Fig. 5.

Profile obtained with the proposed technique for the sampled grating. The pitch is 300 lines/mm.

Fig. 6.
Fig. 6.

Profile obtained with the atomic force microscope in a near vicinity of the measure shown in Fig. 5 when scanning a similar distance.

Equations (31)

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Ψ ( x 0 , y 0 ) = ( 2 P 0 π r 0 2 ) 1 2 exp [ x 0 2 + y 0 2 r 0 2 ] ,
Ψ ( x , y ) = exp ( i ω l t ) 1 iλz Ψ ( x 0 , y 0 ) exp { i π λz [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } dx 0 dy 0 ,
Ψ ( x , y ) = r 0 2 λz r 0 2 2 P 0 π r 0 2 exp ( i ω l t ) exp [ ( π 2 r 0 2 iπλz λ 2 z 2 + π 2 r 0 4 ) ( x 2 + y 2 ) ] ,
Ψ ( x 1 , y 1 ) = 1 λf Ψ ( x , y ) exp [ - i 2 π λf ( x x 1 yy 1 ) ] dx dy .
Ψ ( x 1 , y 1 ) = r 0 2 λf 2 P 0 π r 0 2 exp ( i ω l t ) exp [ ( π π r 0 2 + iλz λ 2 f 2 ) ( x 1 2 + y 1 2 ) ] .
Ψ ( x 2 , y 2 ) = 1 z p Ψ ( x 1 , y 1 ) exp { i π λ z p [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] } dx 1 dy 1 .
Ψ ( x 2 , y 2 ) = π r 0 2 f π r 0 2 z p + ( z z p f 2 ) 2 P 0 π r 0 2 exp ( i ω l t ) ×
exp { π πλ f 2 r 0 2 i [ π 2 r 0 4 z p + λ 2 z ( z z p f 2 ) ] λ [ π 2 r 0 4 z p 2 + λ 2 ( z z p f 2 ) 2 ] ( x 2 2 + y 2 2 ) }
Ψ ( x F , y F ) = A 2 P 0 π r 0 2 exp ( i ω l t ) exp [ ( 1 r 2 + i π λR ) ( x F 2 + y F 2 ) ] ,
A = r 0 2 π r 0 2 z p + ( z z p f 2 ) × π 2 r 0 4 z p 2 + λ 2 ( z z p f 2 ) 2 π λ f 2 r 0 2 i [ π 2 r 0 4 z p + λ 2 z ( z z p f 2 ) ] ,
r = π 2 λ 2 f 4 r 0 4 + [ π 2 r 0 4 z p + λ 2 z ( z z p f 2 ) ] 2 [ π 2 r 0 4 z p 2 + λ 2 ( z z p f 2 ) 2 ] π 2 r 0 2 ,
R = f 2 × π 2 λ 2 f 4 r 0 4 + [ π 2 r 0 4 z p + λ 2 z ( z z p f 2 ) ] 2 [ π 2 r 0 4 z p 2 + λ 2 ( z z p f 2 ) 2 ] [ π 2 r 0 4 z p + λ 2 z ( z z p f 2 ) ]
Ψ p ( ξ , η ) = B 2 P 0 π r 0 2 exp ( i ω l t ) exp [ ( 1 r p 2 i π λ R p ) ( ξ 2 + η 2 ) ] ,
B = A × r 2 R z 2 + r 2 ( z 2 R )
r p = ( R λz 2 ) 2 + π 2 r 4 ( z 2 R ) 2 π 2 r 2 R 2
R p = ( R λz 2 ) 2 + π 2 r 4 ( z 2 R ) 2 ( ) 2 z 2 + π 2 r 4 ( z 2 R ) 2 .
Ψ m ( ξ , η ) = C 2 P 0 π r 0 2 exp [ i ( ω l + ω s ) t ] exp [ ( 1 r m 2 i π λ R m ) ( ξ 2 + η 2 ) ] .
C = r 0 2 λ z 3 r 0 2
r m = λ 2 z 3 2 + π 2 r 0 4 π 2 r 0 2
R m = ( λ 2 z 3 2 + π 2 r 0 4 ) λ 2 z 3 .
Ψ T ( ξ , η ) = Ψ p ( ξ , η ) + Ψ m ( ξ , η ) ,
I ( ξ , η ) = Ψ T ( ξ , η ) Ψ T * ( ξ , η ) ,
I ξ η = D B 2 exp [ 2 r p 2 ( ξ 2 + η 2 ) ] + D C 2 exp [ 2 r m 2 ( ξ 2 + η 2 ) ] +
BC * D exp [ i ( ω s t ) ] exp [ ( α + ) ] [ ξ 2 + η 2 ] +
B * CD exp [ i ( ω s t ) ] exp [ ( α ) ] [ ξ 2 + η 2 ]
D = 2 P 0 π r 0 2 ,
α = r m 2 + r p 2 r m 2 r p 2 ,
β = π λ R m R p ( R p R m ) ,
P ( z p ) = I ξ η dξdη .
P = D B 2 π r p 2 2 + D C 2 π r m 2 2 + B C * D π ( α ) α 2 + β 2 exp ( i ω s t ) + B * CD π ( α + ) α 2 + β 2 exp ( i ω s t ) .
P = P DC + P AC cos ( ω s t + φ ) .

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