Abstract

The combination of a high-speed TV holography system and a 3D Fourier-transform data processing is proposed for the analysis of multimode vibrations in plates. The out-of-plane displacement of the object under generic vibrational excitation is resolved in time by the fast acquisition rate of a high-speed camera, and recorded in a sequence of interferograms with spatial carrier. A full-field temporal history of the multimode vibration is thus obtained. The optical phase of the interferograms is extracted and subtracted from the phase of a reference state to yield a sequence of optical phase-change maps. Each map represents the change undergone by the object between any given state and the reference state. The sequence of maps is a 3D array of data (two spatial dimensions plus time) that is processed with a 3D Fourier-transform algorithm. The individual vibration modes are separated in the 3D frequency space due to their different vibration frequencies and, to a lesser extent, to the different spatial frequencies of the mode shapes. The contribution of each individual mode (or indeed the superposition of several modes) to the dynamic behaviour of the object can then be separated by means of a bandpass filter (or filters). The final output is a sequence of complex-valued maps that contain the full-field temporal history of the selected mode (or modes) in terms of its mechanical amplitude and phase. The proof-of-principle of the technique is demonstrated with a rectangular, fully clamped, thin metal plate vibrating simultaneously in several of its natural resonant frequencies under white-noise excitation.

© 2009 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000).
    [CrossRef]
  2. D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
    [CrossRef]
  3. F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
    [CrossRef]
  4. O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994).
    [CrossRef] [PubMed]
  5. A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
    [CrossRef]
  6. J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
    [CrossRef]
  7. A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
    [CrossRef]
  8. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
    [CrossRef]
  9. C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
    [CrossRef]
  10. W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.
  11. H. O. Saldner, N. E. Molin, and K. A. Stetson, “Fourier transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996).
    [CrossRef] [PubMed]
  12. C. Trillo, and Á. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).
  13. H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in Numerical Recipes in C, (Cambridge University Press 1988).
  14. J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
    [CrossRef] [PubMed]
  15. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), Chap. 8.

2008

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

2000

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
[CrossRef]

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000).
[CrossRef]

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

1999

1997

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

1996

1995

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

1994

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
[CrossRef]

O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994).
[CrossRef] [PubMed]

Barton, J. S.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[CrossRef]

Buckberry, C.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

Buckberry, C. H.

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

Deán, J. L.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

Doval, Á. F.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000).
[CrossRef]

Fernández, J. L.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

Fröning, Ph.

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Ganesan, A. R.

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
[CrossRef]

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
[CrossRef]

Hand, D. P.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[CrossRef]

Hinsch, K. D.

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
[CrossRef]

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
[CrossRef]

Jones, J. D. C.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[CrossRef]

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

Jones, M. P.

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

Kulla, P. H

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Løkberg, O. J.

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994).
[CrossRef] [PubMed]

Meinlschmidt, P.

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
[CrossRef]

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
[CrossRef]

Molin, N. E.

Moore, A. J.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[CrossRef]

Pedersen, H. M.

Pedrini, G.

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Reeves, M.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

Saldner, H. O.

Santoyo, F. M.

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Stetson, K. A.

Stockley, B. C.

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

Tiziani, H. J.

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Towers, D. P.

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

Trillo, C.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

Valera, J. D. R.

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

Valø, H.

Wang, G.

Appl. Opt.

J. Acoust. Soc. Am.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008).
[CrossRef] [PubMed]

Meas. Sci. Technol.

Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000).
[CrossRef]

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995).
[CrossRef]

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997).
[CrossRef]

Opt. Commun.

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000).
[CrossRef]

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994).
[CrossRef]

Opt. Lasers Eng.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000).
[CrossRef]

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000).
[CrossRef]

Other

R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), Chap. 8.

C. Trillo, and Á. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).

H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in Numerical Recipes in C, (Cambridge University Press 1988).

W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.

Supplementary Material (4)

» Media 1: AVI (2547 KB)     
» Media 2: AVI (2613 KB)     
» Media 3: AVI (2606 KB)     
» Media 4: AVI (2526 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Scheme of the reference system and nomenclature of some vibration modes. The dotted lines represent nodal lines.

Fig. 2
Fig. 2

Scheme of the spatiotemporal 3D Fourier transform method. (i) Sequence of optical phase-change maps of a multimode vibration. (ii) Several planes of the spatiotemporal 3D spectrum. The spectral content of one mode (with temporal frequency fk ) of the multimode vibration is shown in planes ±fk . Planes with the spectral content of other modes are omitted. Planes at temporal frequencies f0 and fmax do not contain information. The filter used to select one of the side lobes is also shown. (iii) Mechanical amplitude and phase of the selected mode (modulus and argument of the complex data respectively).

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

(i) Modulus of several complex-valued spectra obtained after processing N=64 optical phase-change maps with the spatiotemporal 3D Fourier transform. The images were equalized and only a small region of 225 pixel × 225 pixel around the center of the spectra is shown. Black and white represent zero and the maximum modulus respectively. A filter is shown on plane n’=14. (ii) 3D representation of a smaller region of 25 pixel×25 pixel.

Fig. 5
Fig. 5

Maximum modulus of the spectra versus n’. The values of temporal frequency (computed for Δt=(1/3560) s, N=64) for the main peaks are indicated. They correspond to M00, M01, M20 and M02.

Fig. 6
Fig. 6

Instantaneous displacement of (a) mode M00, (b) mode M01, (c) mode M20 and (d) mode M02. (e) Operating deflection shape (ODS) due to modes M00, M01, M20 and M02.

Fig. 7
Fig. 7

Instantaneous displacement of (i) mode M00 (Media 1), (ii) mode M01 (Media 2) and (iii) mode M20. (Media 3). (iv): Instantaneous displacement due to the superposition of modes M00, M01 and M20 (Media 4).

Tables (1)

Tables Icon

Table 1 Summary of results

Equations (17)

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

u3=l=0Lm=0Mu3e,lm(x1,x2,l,m)sin[(l+1)πax1]sin[(m+1)πbx2]cos(ωlmt)
ωlm=π2Dρh[(l+1)2a2+(m+1)2b2],D=Eh312(1ν2)
In=gI0,n{1+Vncos(ψp,n+ϕo,nϕr,n+2πfcx)}
ϕo,n=4πλu3
ϕo,n=4πλl=0Lm=0Mu3e,lmsin[(l+1)πax1]sin[(m+1)πbx2]cos(ωlmtn)=l=0Lm=0Mϕ3e,lmcos(ωlmtn)
ΔΦn=ϕo,nϕo,ref=l=0Lm=0M[ϕ3e,lmcos(ωlmtn)ϕ3e,lmcos(ωlmt0)]
ΔΦ(x1p,x2q,tn,l,m)=ΔΦ(x10+pΔx1,x20+qΔx2,t0+nΔt,l,m)
fp'=p'PΔx1,fq'=q'QΔx2,fn'=n'NΔt
ΔΦ'=DFT(ΔΦ)=p=0P1q=0Q1n=0N1{l=1Lm=1M[ϕ3e,lmcos(ωlmtn)ϕ3e,lmcos(ωlmt0)]}×exp(j2πpp'/P)exp(j2πqq'/Q)exp(j2πnn'/N)
ΔΦ'lm=p=0P1q=0Q1ϕ3e,lmexp(j2πpp'/P)exp(j2πqq'/Q)n=0Ncos(ωlmtn)exp(j2πnn'/N)p=0P1q=0Q1ϕ3e,lmexp(j2πpp'/P)exp(j2πqq'/Q)n=0Ncos(ωlmt0)exp(j2πnn'/N)
ΔΦ'lm=DFT(ϕ3e,lm)DFT[cos(2πflmtn)]DFT(ϕ3e,lm)DFT[cos(2πflmt0)]=ϕ'3e,lm2[δ(n'n'lm)+δ(n'+n'lm)]ϕ'3e,lmδ(n')
ΔΦ'lmF=ϕ'3e,lm2δ(n'n'lm).
Αlm(x1p,x2q,tn,l,m)=DFT1(ΔΦ'lmF)=12ϕ3e,lm(x1p,x2q,l,m)exp(j2πflmtn)
mod(Αlm)=Re2(Alm)+Im2(Alm)=ϕ3e,lm2=2πλu3e,lmsin[(l+1)πax1p]sin[(m+1)πbx2p]
arg(Αlm)=arctanIm(Alm)Re(Alm)=2πflmtn
Re(Αlm)=2πλu3e,lmsin[(l+1)πax1p]sin[(m+1)πbx2p]cos(2πflmt)
A(x1p,x2q,tn,l,m)=l=0Lm=0M12ϕ3e,lm(x1p,x2q,l,m)exp(j2πflmtn)

Metrics