Abstract

We describe a closed-form approach for performing a Kramers–Kronig (KK) transform that can be used to rapidly and reliably retrieve the phase, and thus the resonant imaginary component, from a broadband coherent anti-Stokes Raman scattering (CARS) spectrum with a nonflat background. In this approach we transform the frequency-domain data to the time domain, perform an operation that ensures a causality criterion is met, then transform back to the frequency domain. The fact that this method handles causality in the time domain allows us to conveniently account for spectrally varying nonresonant background from CARS as a response function with a finite rise time. A phase error accompanies KK transform of data with finite frequency range. In examples shown here, that phase error leads to small (<1%) errors in the retrieved resonant spectra.

© 2009 U.S. Government

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Rinia, M. Bonn, and M. Muller, J. Phys. Chem. B 110, 4472 (2006).
    [CrossRef] [PubMed]
  2. D. Y. Smith, J. Opt. Soc. Am. 67, 570 (1977).
    [CrossRef]
  3. K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. Vartiainen, Appl. Spectrosc. 58(5), 499 (2004).
    [CrossRef] [PubMed]
  4. D. E. Aspnes, The Accurate Determination of Optical Properties by Ellipsometry, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.
  5. V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).
  6. R. K. Ahrenkiel, J. Opt. Soc. Am. 61, 1651 (1971).
    [CrossRef]
  7. K. F. Palmer, M. Z. Williams, and B. A. Budde, Appl. Opt. 37, 2660 (1998).
    [CrossRef]
  8. C. W. Peterson and B. W. Knight, J. Opt. Soc. Am. 63, 1238 (1973).
    [CrossRef]
  9. D. W. Johnson, J. Phys. A 8, 490 (1975).
    [CrossRef]
  10. E. M. Vartiainen, K. E. Peiponen, and T. Tsuboi, J. Opt. Soc. Am. B 7, 722 (1990).
    [CrossRef]
  11. E. M. Vartiainen, H. A. Rinia, M. Muller, and M. Bonn, Opt. Express 14, 3622 (2006).
    [CrossRef] [PubMed]
  12. Y. X. Liu, Y. J. Lee, and M. T. Cicerone, J. Raman Spectrosc. (to be published).
  13. S. H. Lim, A. G. Caster, and S. R. Leone, Opt. Lett. 32, 1332 (2007).
    [CrossRef] [PubMed]
  14. L. Lepetit, G. Cheriaux, and M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
    [CrossRef]
  15. B. C. Chen and S. H. Lim, J. Phys. Chem. B 112, 3653 (2008).
    [CrossRef] [PubMed]
  16. G. B. Arfken, Mathematical Methods for Physics, 5th ed. (Academic, 2001).
  17. K. Ohta and H. Ishida, Appl. Spectrosc. 42, 952 (1988).
    [CrossRef]
  18. E. M. Vartiainen, J. Opt. Soc. Am. B 9, 1209 (1992).
    [CrossRef]
  19. T. W. Kee, H. X. Zhao, and M. T. Cicerone, Opt. Express 14, 3631 (2006).
    [CrossRef] [PubMed]

2008 (1)

B. C. Chen and S. H. Lim, J. Phys. Chem. B 112, 3653 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (3)

2004 (1)

1998 (1)

1995 (1)

1992 (1)

1990 (1)

1988 (1)

1977 (1)

1975 (1)

D. W. Johnson, J. Phys. A 8, 490 (1975).
[CrossRef]

1973 (1)

1971 (1)

Ahrenkiel, R. K.

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physics, 5th ed. (Academic, 2001).

Aspnes, D. E.

D. E. Aspnes, The Accurate Determination of Optical Properties by Ellipsometry, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.

Bonn, M.

Budde, B. A.

Caster, A. G.

Chen, B. C.

B. C. Chen and S. H. Lim, J. Phys. Chem. B 112, 3653 (2008).
[CrossRef] [PubMed]

Cheriaux, G.

Cicerone, M. T.

T. W. Kee, H. X. Zhao, and M. T. Cicerone, Opt. Express 14, 3631 (2006).
[CrossRef] [PubMed]

Y. X. Liu, Y. J. Lee, and M. T. Cicerone, J. Raman Spectrosc. (to be published).

Ishida, H.

Joffre, M.

Johnson, D. W.

D. W. Johnson, J. Phys. A 8, 490 (1975).
[CrossRef]

Kee, T. W.

Knight, B. W.

Lee, Y. J.

Y. X. Liu, Y. J. Lee, and M. T. Cicerone, J. Raman Spectrosc. (to be published).

Leone, S. R.

Lepetit, L.

Lim, S. H.

Liu, Y. X.

Y. X. Liu, Y. J. Lee, and M. T. Cicerone, J. Raman Spectrosc. (to be published).

Lucarini, V.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. Vartiainen, Appl. Spectrosc. 58(5), 499 (2004).
[CrossRef] [PubMed]

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Muller, M.

Ohta, K.

Palmer, K. F.

Peiponen, K. E.

E. M. Vartiainen, K. E. Peiponen, and T. Tsuboi, J. Opt. Soc. Am. B 7, 722 (1990).
[CrossRef]

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Peiponen, K.-E.

Peterson, C. W.

Rinia, H. A.

Saarinen, J. J.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. Vartiainen, Appl. Spectrosc. 58(5), 499 (2004).
[CrossRef] [PubMed]

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Smith, D. Y.

Tsuboi, T.

Vartiainen, E.

Vartiainen, E. M.

Williams, M. Z.

Zhao, H. X.

Appl. Opt. (1)

Appl. Spectrosc. (2)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (3)

J. Phys. A (1)

D. W. Johnson, J. Phys. A 8, 490 (1975).
[CrossRef]

J. Phys. Chem. B (2)

H. A. Rinia, M. Bonn, and M. Muller, J. Phys. Chem. B 110, 4472 (2006).
[CrossRef] [PubMed]

B. C. Chen and S. H. Lim, J. Phys. Chem. B 112, 3653 (2008).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Other (4)

G. B. Arfken, Mathematical Methods for Physics, 5th ed. (Academic, 2001).

Y. X. Liu, Y. J. Lee, and M. T. Cicerone, J. Raman Spectrosc. (to be published).

D. E. Aspnes, The Accurate Determination of Optical Properties by Ellipsometry, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

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 (2)

Fig. 1
Fig. 1

(a) Numerical simulation of a CARS spectrum via a coherent addition of multiple Lorentzian lineshape functions and a variable nonresonant background (dashed). (b) Temporal function η ( t : ln χ ( ω ) ) , obtained by modified time-domain constraint shown in Eq. (7). Inset, temporal function u ( t ) I 1 [ f ( ω ) ] , obtained without NRB information. (c) (top) Extracted Raman spectrum (solid) and the reference Raman spectrum (dashed); (bottom) difference between the reference and retrieved Raman spectra. Inset, Raman spectrum extracted using u ( t ) I 1 [ f ( ω ) ] from the inset of panel (b).

Fig. 2
Fig. 2

(a) Experimental CARS spectrum of benzonitrile in ethanol at a concentration of 1 M (solid) and a separately measured nonresonant background (dotted). (b) Raman spectrum extracted using Eqs. (6) and (7) and the separately measured background. Uncertainty in the frequency calibration is ± 3 cm 1 .

Equations (7)

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

χ ( ω ) 2 = χ NR 2 + 2 χ NR Re [ χ R ( ω ) ] + χ R ( ω ) 2 .
φ ( ω ) = P π + ln χ ( ω ) ω ω d ω ,
ψ ( f ( ω ) ) = I [ u ( t ) I 1 [ f ( ω ) ] ] ,
ψ ( f ( ω ) ) = 1 2 π I [ u ( t ) ] I [ I 1 [ f ( ω ) ] ] = 1 2 π I [ u ( t ) ] f ( ω ) ,
ψ ( f ( ω ) ) = 1 2 [ i π P + f ( ω ) ω ω d ω + f ( ω ) ] .
φ ( ω ) = 1 π P + ln χ ( ω ) ω ω d ω = 2 Im { ψ ( ln χ ( ω ) ) ln χ ( ω ) 2 } .
η ( t : f ( ω ) ) = { I 1 [ f ( ω ) ] , t 0 I 1 [ f NR ( ω ) ] , t < 0 } ,

Metrics