Abstract

We introduce the concept of phase-synchronous undersampling in nonlinear spectroscopy. The respective theory is presented and validated experimentally in a phase-modulated quantum beat experiment by sampling high phase modulation frequencies with low laser repetition rates. The advantage of undersampling in terms of signal quality and reduced acquisition time is demonstrated, and breakdown conditions are identified. The presented method is particularly beneficial for experimental setups with limited signal/detection rates.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Cho, Chem. Rev. 108, 1331 (2008).
    [Crossref]
  2. P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
    [Crossref]
  3. F. D. Fuller and J. P. Ogilvie, Annu. Rev. Phys. Chem. 66, 667 (2015).
    [Crossref]
  4. P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
    [Crossref]
  5. L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
    [Crossref]
  6. L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
    [Crossref]
  7. P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
    [Crossref]
  8. J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
    [Crossref]
  9. G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, Opt. Express 21, 28617 (2013).
    [Crossref]
  10. K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
    [Crossref]
  11. H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
    [Crossref]
  12. A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
    [Crossref]
  13. L. Bruder, U. Bangert, and F. Stienkemeier, Opt. Express 25, 5302 (2017).
    [Crossref]
  14. O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and Use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).
  15. A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
    [Crossref]
  16. M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
    [Crossref]
  17. C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
    [Crossref]
  18. J. C. Krapez, Proceedings of the Eurotherm Seminar (1998), Vol. 60, pp. 7–10.
  19. P. Tian and W. S. Warren, Opt. Lett. 27, 1634 (2002).
    [Crossref]
  20. K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
    [Crossref]
  21. T. Ye, D. Fu, and W. S. Warren, Photochem. Photobiol. 85, 631 (2009).
    [Crossref]
  22. N. M. Kearns, R. D. Mehlenbacher, A. C. Jones, and M. T. Zanni, Opt. Express 25, 7869 (2017).
    [Crossref]
  23. W. Wagner, C. Li, J. Semmlow, and W. S. Warren, Opt. Express 13, 3697 (2005).
    [Crossref]
  24. S. Draeger, S. Roeding, and T. Brixner, Opt. Express 25, 3259 (2017).
    [Crossref]

2017 (3)

2016 (3)

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
[Crossref]

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

2015 (4)

P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
[Crossref]

F. D. Fuller and J. P. Ogilvie, Annu. Rev. Phys. Chem. 66, 667 (2015).
[Crossref]

L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
[Crossref]

L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
[Crossref]

2014 (1)

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

2013 (3)

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, Opt. Express 21, 28617 (2013).
[Crossref]

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

2009 (1)

T. Ye, D. Fu, and W. S. Warren, Photochem. Photobiol. 85, 631 (2009).
[Crossref]

2008 (2)

M. Cho, Chem. Rev. 108, 1331 (2008).
[Crossref]

M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
[Crossref]

2007 (1)

P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
[Crossref]

2006 (1)

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
[Crossref]

2005 (1)

2002 (1)

2000 (1)

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Autry, T. M.

Bakulin, A. A.

A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
[Crossref]

Bangert, U.

Binz, M.

L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
[Crossref]

Bittner, E. R.

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Bonetto, F.

M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
[Crossref]

Breitenstein, O.

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and Use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

Brixner, T.

S. Draeger, S. Roeding, and T. Brixner, Opt. Express 25, 3259 (2017).
[Crossref]

P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
[Crossref]

Bruder, L.

L. Bruder, U. Bangert, and F. Stienkemeier, Opt. Express 25, 5302 (2017).
[Crossref]

L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
[Crossref]

L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
[Crossref]

Cho, M.

M. Cho, Chem. Rev. 108, 1331 (2008).
[Crossref]

Cundiff, S. T.

Draeger, S.

Dyke, T. R.

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
[Crossref]

Fu, D.

T. Ye, D. Fu, and W. S. Warren, Photochem. Photobiol. 85, 631 (2009).
[Crossref]

Fuller, F. D.

F. D. Fuller and J. P. Ogilvie, Annu. Rev. Phys. Chem. 66, 667 (2015).
[Crossref]

Gauthier-Houle, A.

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Gregoire, P.

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Hartov, A.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Johnson, N. P.

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

Jones, A. C.

Jose, D.

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

Karki, K. J.

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

Kearns, N. M.

Kerner, T. E.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Krapez, J. C.

J. C. Krapez, Proceedings of the Eurotherm Seminar (1998), Vol. 60, pp. 7–10.

Kringle, L.

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

Langenkamp, M.

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and Use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

Lee, W.

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

Li, C.

Li, H.

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Lonergan, M. C.

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

Lott, G. A.

P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
[Crossref]

Marcus, A. H.

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
[Crossref]

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
[Crossref]

Mazzarese, R. A.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Mehlenbacher, R. D.

Moody, I.

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

Mudrich, M.

L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
[Crossref]

Nardin, G.

Nuernberger, P.

P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
[Crossref]

Ogilvie, J. P.

F. D. Fuller and J. P. Ogilvie, Annu. Rev. Phys. Chem. 66, 667 (2015).
[Crossref]

Osterman, K. S.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Paulsen, K. D.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Phelps, C.

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

Pullerits, T.

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

Reiss, F. R.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Roeding, S.

Ruetzel, S.

P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
[Crossref]

Seibt, J.

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

Semmlow, J.

Silva, C.

A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
[Crossref]

Silva-Acuña, C.

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Silverman, K. L.

Sonnaillon, M.

M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
[Crossref]

Stienkemeier, F.

L. Bruder, U. Bangert, and F. Stienkemeier, Opt. Express 25, 5302 (2017).
[Crossref]

L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
[Crossref]

L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
[Crossref]

Tekavec, P. F.

P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
[Crossref]

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
[Crossref]

Tian, P.

Urteaga, R.

M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
[Crossref]

Vella, E.

A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
[Crossref]

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

von Hippel, P. H.

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

Wagner, W.

Warren, W. S.

Widom, J. R.

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

Williams, D. B.

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

Ye, T.

T. Ye, D. Fu, and W. S. Warren, Photochem. Photobiol. 85, 631 (2009).
[Crossref]

Zanni, M. T.

Angew. Chem. Int. Ed. (1)

P. Nuernberger, S. Ruetzel, and T. Brixner, Angew. Chem. Int. Ed. 54, 11368 (2015).
[Crossref]

Annu. Rev. Phys. Chem. (1)

F. D. Fuller and J. P. Ogilvie, Annu. Rev. Phys. Chem. 66, 667 (2015).
[Crossref]

Chem. Phys. (1)

H. Li, A. Gauthier-Houle, P. Gregoire, E. Vella, C. Silva-Acuña, and E. R. Bittner, Chem. Phys. 481, 281 (2016).
[Crossref]

Chem. Rev. (1)

M. Cho, Chem. Rev. 108, 1331 (2008).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

A. Hartov, R. A. Mazzarese, F. R. Reiss, T. E. Kerner, K. S. Osterman, D. B. Williams, and K. D. Paulsen, IEEE Trans. Biomed. Eng. 47, 49 (2000).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

M. Sonnaillon, R. Urteaga, and F. Bonetto, IEEE Trans. Instrum. Meas. 57, 616 (2008).
[Crossref]

J. Chem. Phys. (2)

P. F. Tekavec, G. A. Lott, and A. H. Marcus, J. Chem. Phys. 127, 214307 (2007).
[Crossref]

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, J. Chem. Phys. 125, 194303 (2006).
[Crossref]

J. Opt. (1)

K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. 18, 015504 (2016).
[Crossref]

J. Phys. Chem. Lett. (1)

A. A. Bakulin, C. Silva, and E. Vella, J. Phys. Chem. Lett. 7, 250 (2016).
[Crossref]

Nat. Commun. (1)

K. J. Karki, J. R. Widom, J. Seibt, I. Moody, M. C. Lonergan, T. Pullerits, and A. H. Marcus, Nat. Commun. 5, 5869 (2014).
[Crossref]

New J. Phys. (1)

J. R. Widom, N. P. Johnson, P. H. von Hippel, and A. H. Marcus, New J. Phys. 15, 025028 (2013).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Photochem. Photobiol. (1)

T. Ye, D. Fu, and W. S. Warren, Photochem. Photobiol. 85, 631 (2009).
[Crossref]

Phys. Chem. Chem. Phys. (1)

L. Bruder, M. Mudrich, and F. Stienkemeier, Phys. Chem. Chem. Phys. 17, 23877 (2015).
[Crossref]

Phys. Rev. A (1)

L. Bruder, M. Binz, and F. Stienkemeier, Phys. Rev. A 92, 053412 (2015).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

C. Phelps, W. Lee, D. Jose, P. H. von Hippel, and A. H. Marcus, Proc. Natl. Acad. Sci. USA 110, 17320 (2013).
[Crossref]

Other (2)

J. C. Krapez, Proceedings of the Eurotherm Seminar (1998), Vol. 60, pp. 7–10.

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and Use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

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Figures (5)

Fig. 1.
Fig. 1. Optical PM setup. The signal S is modulated using two AOMs driven at the frequency difference Ω = Ω 2 Ω 1 . A cw reference signal R is generated with a diode laser. Signal demodulation is done with a LIA, referenced to R . Abbreviations: photodiode (PD), pinhole (PH), beamsplitter (BS), delay line (DL).
Fig. 2.
Fig. 2. S/N advantage by upshifting S from low frequencies to Ω where the noise floor is much smaller and S is not dominated by lab noise.
Fig. 3.
Fig. 3. (a) Schematic description of the lock-in amplification scheme. LP denotes low-pass filter. (b) Representation of the product signal S R , as constructed in the LIA. Green: fully sampled signal ( S R ); black: undersampled signal ( S R us ); dashed black: resulting aliased waveform ( S R a ); and red: mean value of S R us scaled according to the sampling duty cycle ( S R ). Undersampling of S R yields the correct average value/output signal, if the averaging interval T avg is sufficiently large. (c) Optimum undersampling case. (d) Breakdown case.
Fig. 4.
Fig. 4. (a) SNR as a function of the modulation frequency Ω , recorded for a fixed sampling frequency of ω sa = 5    kHz . (b) Comparison of different averaging intervals T avg , varied via the lock-in filter time T LI .
Fig. 5.
Fig. 5. SNR for fully and undersampled signal modulation, measured for laser repetition rates of ω rep = 5 and 0.5 kHz. The latter data set was scaled by a factor of 10 for better visibility. Blue and red shaded areas indicate the fully sampled regime for the two data sets, respectively. Gray: noise spectrum of the lab, as picked up in the experiment.

Equations (7)

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

Ω a = min | n ω sa Ω | 0.5 ω sa , n Z ,
S ( t , τ ) = A ( τ ) cos [ Ω t + ϕ S ( τ ) ] ,
R ( t , τ ) = cos [ Ω t + ϕ R ( τ ) ] ,
S R ( t , τ ) cos [ 2 Ω t + ϕ S ( τ ) + ϕ R ( τ ) ] + cos [ ϕ S R ( τ ) ] ,
S R ( τ ) = 0.5 A ( τ ) cos [ ϕ S R ( τ ) ] .
( i ) :    4 Ω = ( 2 n + 1 ) ω sa Ω a S R = 0.5 ω sa ,
( ii ) :    2 Ω = n ω sa Ω a S R = 0 ,

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