Abstract

A general solution of undersampling frequency conversion and its optimization for parallel photodisplacement imaging is presented. Phase-modulated heterodyne interference light generated by a linear region of periodic displacement is captured by a charge-coupled device image sensor, in which the interference light is sampled at a sampling rate lower than the Nyquist frequency. The frequencies of the components of the light, such as the sideband and carrier (which include photodisplacement and topography information, respectively), are downconverted and sampled simultaneously based on the integration and sampling effects of the sensor. A general solution of frequency and amplitude in this downconversion is derived by Fourier analysis of the sampling procedure. The optimal frequency condition for the heterodyne beat signal, modulation signal, and sensor gate pulse is derived such that undesirable components are eliminated and each information component is converted into an orthogonal function, allowing each to be discretely reproduced from the Fourier coefficients. The optimal frequency parameters that maximize the sideband-to-carrier amplitude ratio are determined, theoretically demonstrating its high selectivity over 80  dB. Preliminary experiments demonstrate that this technique is capable of simultaneous imaging of reflectivity, topography, and photodisplacement for the detection of subsurface lattice defects at a speed corresponding to an acquisition time of only 0 .26  s per 256×256 pixel area.

© 2006 Optical Society of America

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References

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  1. R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
    [CrossRef]
  2. A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
    [CrossRef]
  3. A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
    [CrossRef]
  4. M. C. Jackson and P. Matthewson, "Digital processing of bandpass signals," GEC J. Res. 4, 32-41 (1986).
  5. O. D. Grace and S. P. Pitt, "Quadrature sampling of high frequency waveforms," J. Acoust. Soc. Am. 44, 1432-1436 (1968).
    [CrossRef]
  6. F. J. J. Clarke and J. R. Stockton, "Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs," J. Phys. E 15, 645-652 (1982).
    [CrossRef]
  7. R. Kohno, "Structures and theories of software antennas for software defined radio," IEICE Trans. Commun. E83-B, 1189-1199 (2000).
  8. S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).
  9. K. Tatsuno and Y. Tsunoda, "Diode laser direct modulation heterodyne interferometer," Appl. Opt. 26, 37-40 (1987).
    [CrossRef] [PubMed]
  10. T. Nakata, H. H. Kobayashi, and T. Ninomiya, "Study on high-speed photothermal displacement microscopy using parallel excitation and phase-shifting signal integration," in Proceedings of the Fourteenth Symposium on Ultrasonic Electronics (Japan Society of Applied Physics, Tokyo, 1993), pp. 81-82.
  11. T. Nakata and T. Ninomiya, "Practical realization of high-speed photodisplacement imaging by use of parallel excitation and parallel heterodyne detection: a numerical study," Appl. Opt. 43, 3287-3296 (2004).
    [CrossRef] [PubMed]
  12. T. Nakata and T. Ninomiya, "A charge-coupled-device-based heterodyne technique for parallel photodisplacement imaging," J. Appl. Phys. 96, 6970-6980 (2004).
    [CrossRef]
  13. T. Nakata and T. Ninomiya, "Real-time photodisplacement imaging using parallel excitation and parallel heterodyne interferometry," J. Appl. Phys. 97, 103110 (2005).
    [CrossRef]
  14. T. Nakata and T. Ninomiya, "Simultaneous real-time imaging of surface and subsurface structures from a single space-frequency multiplexed photodisplacement interferogram," Appl. Opt. 44, 5809-5817 (2005).
    [CrossRef] [PubMed]
  15. T. Nakata, K. Yoshimura, and T. Ninomiya, "Real-time photodisplacement microscope for high-sensitivity simultaneous surface and subsurface inspection," Appl. Opt. 45, 2643-2655 (2006).
    [CrossRef] [PubMed]
  16. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), Chap. 10.
  17. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), Chap. 2.
  18. S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
    [CrossRef]
  19. Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
    [CrossRef]

2006 (1)

2005 (2)

T. Nakata and T. Ninomiya, "Real-time photodisplacement imaging using parallel excitation and parallel heterodyne interferometry," J. Appl. Phys. 97, 103110 (2005).
[CrossRef]

T. Nakata and T. Ninomiya, "Simultaneous real-time imaging of surface and subsurface structures from a single space-frequency multiplexed photodisplacement interferogram," Appl. Opt. 44, 5809-5817 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (1)

S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).

2000 (1)

R. Kohno, "Structures and theories of software antennas for software defined radio," IEICE Trans. Commun. E83-B, 1189-1199 (2000).

1995 (1)

A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
[CrossRef]

1994 (2)

A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
[CrossRef]

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

1992 (1)

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

1991 (1)

R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
[CrossRef]

1987 (1)

1986 (1)

M. C. Jackson and P. Matthewson, "Digital processing of bandpass signals," GEC J. Res. 4, 32-41 (1986).

1982 (1)

F. J. J. Clarke and J. R. Stockton, "Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs," J. Phys. E 15, 645-652 (1982).
[CrossRef]

1968 (1)

O. D. Grace and S. P. Pitt, "Quadrature sampling of high frequency waveforms," J. Acoust. Soc. Am. 44, 1432-1436 (1968).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), Chap. 10.

Champeney, D. C.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), Chap. 2.

Clarke, F. J. J.

F. J. J. Clarke and J. R. Stockton, "Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs," J. Phys. E 15, 645-652 (1982).
[CrossRef]

Coulson, A. J.

A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
[CrossRef]

A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
[CrossRef]

Grace, O. D.

O. D. Grace and S. P. Pitt, "Quadrature sampling of high frequency waveforms," J. Acoust. Soc. Am. 44, 1432-1436 (1968).
[CrossRef]

Horiuchi, T.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Jackson, M. C.

M. C. Jackson and P. Matthewson, "Digital processing of bandpass signals," GEC J. Res. 4, 32-41 (1986).

Karasawa, Y.

S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).

Kawata, Y.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Kita, T.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Kobayashi, H. H.

T. Nakata, H. H. Kobayashi, and T. Ninomiya, "Study on high-speed photothermal displacement microscopy using parallel excitation and phase-shifting signal integration," in Proceedings of the Fourteenth Symposium on Ultrasonic Electronics (Japan Society of Applied Physics, Tokyo, 1993), pp. 81-82.

Koda, T.

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Kohno, R.

R. Kohno, "Structures and theories of software antennas for software defined radio," IEICE Trans. Commun. E83-B, 1189-1199 (2000).

Matthewson, P.

M. C. Jackson and P. Matthewson, "Digital processing of bandpass signals," GEC J. Res. 4, 32-41 (1986).

Morimoto, T.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Nagata, Y.

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Nakano, S.

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Nakata, T.

T. Nakata, K. Yoshimura, and T. Ninomiya, "Real-time photodisplacement microscope for high-sensitivity simultaneous surface and subsurface inspection," Appl. Opt. 45, 2643-2655 (2006).
[CrossRef] [PubMed]

T. Nakata and T. Ninomiya, "Simultaneous real-time imaging of surface and subsurface structures from a single space-frequency multiplexed photodisplacement interferogram," Appl. Opt. 44, 5809-5817 (2005).
[CrossRef] [PubMed]

T. Nakata and T. Ninomiya, "Real-time photodisplacement imaging using parallel excitation and parallel heterodyne interferometry," J. Appl. Phys. 97, 103110 (2005).
[CrossRef]

T. Nakata and T. Ninomiya, "A charge-coupled-device-based heterodyne technique for parallel photodisplacement imaging," J. Appl. Phys. 96, 6970-6980 (2004).
[CrossRef]

T. Nakata and T. Ninomiya, "Practical realization of high-speed photodisplacement imaging by use of parallel excitation and parallel heterodyne detection: a numerical study," Appl. Opt. 43, 3287-3296 (2004).
[CrossRef] [PubMed]

T. Nakata, H. H. Kobayashi, and T. Ninomiya, "Study on high-speed photothermal displacement microscopy using parallel excitation and phase-shifting signal integration," in Proceedings of the Fourteenth Symposium on Ultrasonic Electronics (Japan Society of Applied Physics, Tokyo, 1993), pp. 81-82.

Nakayama, H.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Ninomiya, T.

T. Nakata, K. Yoshimura, and T. Ninomiya, "Real-time photodisplacement microscope for high-sensitivity simultaneous surface and subsurface inspection," Appl. Opt. 45, 2643-2655 (2006).
[CrossRef] [PubMed]

T. Nakata and T. Ninomiya, "Real-time photodisplacement imaging using parallel excitation and parallel heterodyne interferometry," J. Appl. Phys. 97, 103110 (2005).
[CrossRef]

T. Nakata and T. Ninomiya, "Simultaneous real-time imaging of surface and subsurface structures from a single space-frequency multiplexed photodisplacement interferogram," Appl. Opt. 44, 5809-5817 (2005).
[CrossRef] [PubMed]

T. Nakata and T. Ninomiya, "A charge-coupled-device-based heterodyne technique for parallel photodisplacement imaging," J. Appl. Phys. 96, 6970-6980 (2004).
[CrossRef]

T. Nakata and T. Ninomiya, "Practical realization of high-speed photodisplacement imaging by use of parallel excitation and parallel heterodyne detection: a numerical study," Appl. Opt. 43, 3287-3296 (2004).
[CrossRef] [PubMed]

T. Nakata, H. H. Kobayashi, and T. Ninomiya, "Study on high-speed photothermal displacement microscopy using parallel excitation and phase-shifting signal integration," in Proceedings of the Fourteenth Symposium on Ultrasonic Electronics (Japan Society of Applied Physics, Tokyo, 1993), pp. 81-82.

Nishimoto, Y.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Nishino, T.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Ogiso, H.

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Pitt, S. P.

O. D. Grace and S. P. Pitt, "Quadrature sampling of high frequency waveforms," J. Acoust. Soc. Am. 44, 1432-1436 (1968).
[CrossRef]

Poletti, M. A.

A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
[CrossRef]

Sasaki, S.

S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).

Scott, N. L.

A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
[CrossRef]

R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
[CrossRef]

Stockton, J. R.

F. J. J. Clarke and J. R. Stockton, "Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs," J. Phys. E 15, 645-652 (1982).
[CrossRef]

Sumie, S.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Takamatsu, H.

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

Taniguchi, T.

S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).

Tatsuno, K.

Tsunoda, Y.

Vaughan, R. G.

A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
[CrossRef]

A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
[CrossRef]

R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
[CrossRef]

White, D. R.

R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
[CrossRef]

Yamanaka, K.

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Yoshimura, K.

Appl. Opt. (4)

GEC J. Res. (1)

M. C. Jackson and P. Matthewson, "Digital processing of bandpass signals," GEC J. Res. 4, 32-41 (1986).

IEEE Trans. Signal Process. (3)

R. G. Vaughan, N. L. Scott, and D. R. White, "The theory of bandpass sampling," IEEE Trans. Signal Process. 39, 1973-1984 (1991).
[CrossRef]

A. J. Coulson, R. G. Vaughan, and M. A. Poletti, "Frequency shifting using bandpass sampling," IEEE Trans. Signal Process. 42, 1556-1559 (1994).
[CrossRef]

A. J. Coulson, R. G. Vaughan, and N. L. Scott, "Signal combination using bandpass sampling," IEEE Trans. Signal Process. 43, 1809-1818 (1995).
[CrossRef]

IEICE Trans. Commun. (2)

R. Kohno, "Structures and theories of software antennas for software defined radio," IEICE Trans. Commun. E83-B, 1189-1199 (2000).

S. Sasaki, T. Taniguchi, and Y. Karasawa, "An adaptive array antenna based on the IQ-division bandpass sampling," IEICE Trans. Commun. E86-B, 3483-3490 (2003).

J. Acoust. Soc. Am. (1)

O. D. Grace and S. P. Pitt, "Quadrature sampling of high frequency waveforms," J. Acoust. Soc. Am. 44, 1432-1436 (1968).
[CrossRef]

J. Appl. Phys. (3)

T. Nakata and T. Ninomiya, "A charge-coupled-device-based heterodyne technique for parallel photodisplacement imaging," J. Appl. Phys. 96, 6970-6980 (2004).
[CrossRef]

T. Nakata and T. Ninomiya, "Real-time photodisplacement imaging using parallel excitation and parallel heterodyne interferometry," J. Appl. Phys. 97, 103110 (2005).
[CrossRef]

S. Sumie, H. Takamatsu, T. Morimoto, Y. Nishimoto, Y. Kawata, T. Horiuchi, H. Nakayama, T. Kita, and T. Nishino, "Analysis of lattice defects induced by ion implantation with photoacoustic displacement measurements," J. Appl. Phys. 76, 5681-5689 (1994).
[CrossRef]

J. Phys. E (1)

F. J. J. Clarke and J. R. Stockton, "Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs," J. Phys. E 15, 645-652 (1982).
[CrossRef]

Nondestr. Test. Eval. (1)

Y. Nagata, K. Yamanaka, H. Ogiso, S. Nakano, and T. Koda, "Characterization of ion implanted silicon and diamond by variable wavelength photoacoustic microscopy and scanning acoustic microscopy," Nondestr. Test. Eval. 8-9, 1013-1023 (1992).
[CrossRef]

Other (3)

T. Nakata, H. H. Kobayashi, and T. Ninomiya, "Study on high-speed photothermal displacement microscopy using parallel excitation and phase-shifting signal integration," in Proceedings of the Fourteenth Symposium on Ultrasonic Electronics (Japan Society of Applied Physics, Tokyo, 1993), pp. 81-82.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), Chap. 10.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), Chap. 2.

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Figures (12)

Fig. 1
Fig. 1

Parallel excitation and parallel heterodyne detection.

Fig. 2
Fig. 2

Integration and readout procedure in the CCD sensor.

Fig. 3
Fig. 3

Undersampling of an input signal with a single frequency.

Fig. 4
Fig. 4

Undersampling procedure in the frequency domain. (a) Spectra of original frequencies f V and f W . (b) Spectrum of undersampled signal E V ( f ) . (c) Spectrum of undersampled signal E W ( f ) .

Fig. 5
Fig. 5

Frequency condition for elimination of sideband at f B + f E .

Fig. 6
Fig. 6

Frequency condition for elimination of sideband at f B f E .

Fig. 7
Fig. 7

Frequency condition for sampling of downconverted sideband at the rate of π / 2 .

Fig. 8
Fig. 8

Generation of control signals by phase-locked loop circuit.

Fig. 9
Fig. 9

Dependences of carrier and sideband amplitude on frequency parameters. (a) Dependence of carrier amplitude on parameter u. (b) Dependence of sideband amplitude on parameter | u v | .

Fig. 10
Fig. 10

Theoretical sideband-to-carrier amplitude ratio. (a) Conventional photodisplacement technique. (b) Parallel photodisplacement technique.

Fig. 11
Fig. 11

Frequency response for elimination of the carrier component, coupled with extraction of the sideband component, where f S = 47.059 kHz, f B = 100 kHz, and f E = 88.235 kHz for p = 1 and u = v = 2 in Eq. (41), for the processing of the sideband component over 12 cycles.

Fig. 12
Fig. 12

Example of surface and subsurface images obtained simultaneously for analysis of subsurface lattice defects. (a) Silicon wafer locally implanted with 300   keV   Ar + ions at a dose of 1 × 10 15 ions / cm 2 used as a subsurface defect sample. (b) Reflectivity image. (c) Topography image. (d) Photodisplacement amplitude image. (e) Photodisplacement phase image.

Tables (2)

Tables Icon

Table 1 General Solution of Undersampling Frequency Conversion a

Tables Icon

Table 2 Examples of Optimal Frequency Condition

Equations (53)

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

I ( x , t ) = 2 α I L R s ( x ) + 2 α I L R s ( x ) × { cos [ 2 π f B t + ϕ path ( x ) + 4 π n a h ( x ) λ ] + 2 π λ A ( x ) sin [ 2 π ( f B + f E ) t + ϕ path ( x ) + 4 π n a h ( x ) λ + θ ( x ) ] + 2 π λ A ( x ) sin [ 2 π ( f B f E ) t + ϕ path ( x ) + 4 π n a h ( x ) λ θ ( x ) ] } ,
S ( m , t ) = t t + Δ t I ( m , t ) d t = 2 α I L R s ( m ) Δ t + 2 α I L R s ( m ) { B 0 2 π f B × cos [ 2 π f B t + Φ 0 + ϕ path ( m ) + 4 π n a h ( m ) λ ] + A ( m ) λ B 1 f B + f E sin [ 2 π ( f B + f E ) t + Φ 1 + ϕ path ( m ) + 4 π n a h ( m ) λ + θ ( m ) ] + A ( m ) λ B 2 f B f E sin [ 2 π ( f B f E ) t + Φ 2 + ϕ path ( m ) + 4 π n a h ( m ) λ θ ( m ) ] } ,
B 0 = 2 sin ( π f B Δ t ) ,
B 1 = 2 sin [ π ( f B + f E ) Δ t ] ,
B 2 = 2 sin [ π ( f B f E ) Δ t ] ,
Φ 0 = π f B Δ t ,
Φ 1 = π ( f B + f E ) Δ t ,
Φ 2 = π ( f B f E ) Δ t .
f S > 2 ( f B + f E ) .
f S < 2 ( f B + f E ) .
2 f U g f S 2 f L g 1 ,
1 g I g [ f U B ] ,
e C ( t ) = q = + e O ( q 1 f S ) δ ( t q 1 f S ) ,
E V ( f ) = p = + 2 π f S { 1 2 i δ [ f q f S + ( f V n f S ) ] 1 2 i δ [ f q f S ( f V n f S ) ] } ,
E W ( f ) = r = + 2 π f S ( 1 2 i δ { f q f S + [ ( n + 1 ) f S f W ] } 1 2 i δ { f q f S [ ( n + 1 ) f S f W ] } ) ,
n = I h [ f O f S ] 0 ,
F B E = f S r ,
F B = f S s ,
f B + f E = m f S ,
f E = n f S + ( f S G B ) ,
f E = n f S + G B + ( f S G B E ) ,
f S G B = G B + ( f S G B E ) ,
G B E = 2 G B .
F B E = f S r ,
F B = f S 2 r .
f S : f B : f E = f S : 2 r m 2 r ( n + 1 ) ± 1 2 r f S : 2 r ( n + 1 ) 1 2 r f S
= 2 r : 2 r ( m n 1 ) ± 1 : 2 r ( n + 1 ) 1
= 2 r : 2 r u ± 1 : 2 r v 1 ,
B 1 = 2 sin [ ( u + v ) π ] = 0 ,
f B f E = m f S .
H B + E = 2 H B ,
F B + E = f S r ,
F B = f S 2 r .
f S : f B : f E = 2 r : 2 r u ± 1 : 2 r v ± 1 ,
S ( m , i ) = a 0 ( m ) 2 + n = 1 N / 2 [ a n ( m ) cos ( 2 π n N i ) + b n ( m ) sin ( 2 π n N i ) ] ,
a n ( m ) = 2 N i = 0 N 1 S ( m , i ) cos ( 2 π n N i ) ,
b n ( m ) = 2 N i = 0 N 1 S ( m , i ) sin ( 2 π n N i ) .
a N / 4 p ( m ) = 2 p N [ S ( m , 0 ) S ( m , 2 p ) + S ( m , 4 p ) S ( m , 6 p ) + ] ,
b N / 4 p ( m ) = 2 p N [ S ( m , p ) S ( m , 3 p ) + S ( m , 5 p ) S ( m , 7 p ) + ] .
r = 4 p .
F B E = f S 4 p ,
F B = f S 8 p ,
f S : f B : f E = 8 p : 8 p u ± 1 : 8 p v 1 ,
F B + E = f S 4 p ,
F B = f S 8 p ,
f S : f B : f E = 8 p : 8 p u ± 1 : 8 pv ± 1 ,
a N / 8 p ( m ) = 4 p N [ S ( m , 0 ) S ( m , 4 p ) + S ( m , 8 p ) S ( m , 12 p ) + ] ,
b N / 8 p ( m ) = 4 p N [ S ( m , 2 p ) S ( m , 6 p ) + S ( m , 10 p ) S ( m , 14 p ) + ] .
a 0 ( m ) = 2 N [ S ( m , 0 ) + S ( m , 1 ) + S ( m , 2 ) + S ( m , 3 ) + ] .
R s ( m ) = f S 2 α I L a 0 ( m ) ,
h ( m ) = λ 4 π n a [ tan 1 b N / 8 p ( m ) a N / 8 p ( m ) π 8 p ϕ path ( m ) ] ,
A ( m ) = λ ( f B f E ) 2 f S sin ( ± π 4 p ) a 0 ( m ) × [ a N / 4 p 2 ( m ) + b N / 4 p 2 ( m ) ] 1 / 2 ,
θ ( m ) = tan 1 b N / 4 p ( m ) a N / 4 p ( m ) + tan 1 b N / 8 p ( m ) a N / 8 p ( m ) ± π 8 p .

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