Abstract

We investigate the controversy regarding violations of the Bouguer–Lambert–Beer (BLB) law for ultrashort laser pulses propagating through water. By working at sufficiently low incident laser intensities, we make sure that any nonlinear component in the response of the medium is negligible. We measure the transmitted power and spectrum as functions of water cell length in an effort to confirm or disprove alleged deviations from the BLB law. We perform experiments at two different laser pulse repetition rates and explore the dependence of transmission on pulse duration. Specifically, we vary the laser pulse duration either by cutting its spectrum while keeping the pulse shape near transform-limited or by adjusting the pulses chirp while keeping the spectral intensities fixed. Over a wide range of parameters, we find no deviations from the BLB law and conclude that recent claims of BLB law violations are inconsistent with our experimental data. We present a simple linear theory (based on the BLB law) for propagation of ultrashort laser pulses through an absorbing medium and find our experimental results to be in excellent agreement with this theory.

© 2009 Optical Society of America

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References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. U. J. Gibson and U. L. Österberg, “Optical precursors and Beer's law violations: nonexponential propagation losses in water,” Opt. Express 13, 2105-2110 (2005).
    [CrossRef] [PubMed]
  9. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A. L. Gaeta, “Absorption of ultrashort optical pulses in water,” J. Opt. Soc. Am. A 24, 3343-3347 (2007).
    [CrossRef]
  10. R. M. Pope and E. S. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710-8723 (1997).
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  11. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5 ?m spectral range,” Appl. Opt. 32, 3531-3540 (1993).
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    [CrossRef]
  13. D. Segelstein, “The complex refractive index of water,” master's thesis (University of Missouri-Kansas City, 1981).
  14. G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media--I Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13, 1065-1080 (1973).
    [CrossRef]

2007 (2)

2006 (1)

2005 (2)

U. J. Gibson and U. L. Österberg, “Optical precursors and Beer's law violations: nonexponential propagation losses in water,” Opt. Express 13, 2105-2110 (2005).
[CrossRef] [PubMed]

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

2004 (2)

T. M. Roberts, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 93, 269401 (2004).
[CrossRef]

S. H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92, 193903 (2004).
[CrossRef] [PubMed]

2000 (1)

1997 (1)

1993 (1)

1988 (1)

1981 (1)

D. Segelstein, “The complex refractive index of water,” master's thesis (University of Missouri-Kansas City, 1981).

1973 (1)

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media--I Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13, 1065-1080 (1973).
[CrossRef]

1960 (1)

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Agha, I. H.

Alexander, D. R.

Alfano, R. R.

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

Alrubaiee, M.

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

Birman, J. L.

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Bruce, J. C.

Choi, S. H.

S. H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92, 193903 (2004).
[CrossRef] [PubMed]

Chylek, P.

Das, B. B.

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

Doerr, D. W.

Fox, A. E.

Fry, E. S.

Gaeta, A. L.

Geraghty, D. F.

Gibson, U. J.

Kattawar, G. W.

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media--I Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13, 1065-1080 (1973).
[CrossRef]

Kou, L.

Labrie, D.

Li, J. C.

Ni, X.

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

Okawachi, Y.

Österberg, U.

Österberg, U. L.

Oughstun, K. E.

Parali, U. P.

Plass, G. N.

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media--I Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13, 1065-1080 (1973).
[CrossRef]

Pope, R. M.

Roberts, T. M.

T. M. Roberts, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 93, 269401 (2004).
[CrossRef]

Segelstein, D.

D. Segelstein, “The complex refractive index of water,” master's thesis (University of Missouri-Kansas City, 1981).

Sherman, G. C.

Slepkov, A. D.

Wang, H.

Zhang, H. F.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media--I Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13, 1065-1080 (1973).
[CrossRef]

Opt. Express (3)

Phys. Rev. Lett. (3)

S. H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92, 193903 (2004).
[CrossRef] [PubMed]

R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 94, 239401 (2005).
[CrossRef] [PubMed]

T. M. Roberts, “Comment on 'Observation of optical precursors in water',” Phys. Rev. Lett. 93, 269401 (2004).
[CrossRef]

Other (2)

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

D. Segelstein, “The complex refractive index of water,” master's thesis (University of Missouri-Kansas City, 1981).

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Figures (15)

Fig. 1
Fig. 1

Simulated total power transmission as a function of propagation distance for pulses with varying center wavelengths and bandwidths. The circles, squares, and triangles denote center wavelengths of 760, 800, and 840 nm , respectively. The solid and dashed lines denote bandwidths of 45 and 90 nm , respectively.

Fig. 2
Fig. 2

Simulated average absorption coefficient α ¯ as a function of propagation distance for pulses with varying center wavelengths and bandwidths. The circles, squares, and triangles denote center wavelengths of 760, 800, and 840 nm , respectively. The solid and dashed lines denote bandwidths of 45 and 90 nm , respectively.

Fig. 3
Fig. 3

Experimental schematic. The beam path through the amplifier, pulse shaper, and telescope is optional, depending on the experiment performed. The upper right inset shows the input pulse spectrum, while the lower right inset shows a frequency-resolved optical gating trace of the transform-limited input pulse. ND, neutral density.

Fig. 4
Fig. 4

Normalized transmitted power for amplified pulses propagating through 150 cm of the water sample as a function of incident power. The triangles designate powermeter measurements, while the circles designate integrated spectrometer measurements. The dashed line represents the expected trend for linear behavior. It can be seen that for total incident powers above 8.6 mW , the transmitted power deviates from linear behavior. Therefore, all experiments are conducted at incident power levels at or below this point. All error bars are smaller than the size of the symbols.

Fig. 5
Fig. 5

Spectra for various incident powers normalized to the input spectrum. P in ( ω 0 ) is the incident spectral power at the center wavelength. In the order of descending spectral power at the center wavelength, the curves from top to bottom correspond to total incident powers of 4.8, 1.6, 0.9, 8.6, and 16 mW , respectively. Normalized curves for total incident powers at and below 8.6 mW resemble each other, while normalized curves with greater total incident power begin to deviate from the input spectrum as a result of nonlinear behavior.

Fig. 6
Fig. 6

Total measured integrated power as a function of propagation distance as measured by the spectrometer. The circles and squares represent amplified pulses with total incident powers of 8.6 and 4.8 mW , respectively. The triangles represent oscillator pulses with total incident power of 43 mW ; solid lines indicate the predicted simulated behavior for each pulse. All error bars are smaller than the size of the symbols.

Fig. 7
Fig. 7

Total measured power as a function of propagation distance as measured by the powermeter. The circles and squares represent amplified pulses with total incident powers of 8.6 and 4.8 mW , respectively. The triangles represent oscillator pulses with total incident power of 43 mW ; solid lines indicate the predicted simulated behavior for each pulse. All error bars are smaller than the size of the symbols.

Fig. 8
Fig. 8

Spectral power for various wavelengths as a function of propagation distance for laser oscillator pulses with total incident power of 43 mW . Squares, circles, and triangles with apices up and down represent wavelengths of 700, 800, 810, and 820 nm , respectively. The solid lines are exponential fits weighted with uncertainty. All error bars are smaller than the size of the symbols.

Fig. 9
Fig. 9

Spectral power for various wavelengths as a function of propagation distance for amplified pulses with total incident power of 4.8 mW . Squares, circles, and triangles with apices up and down represent wavelengths of 700, 800, 810, and 820 nm , respectively. The solid lines are exponential fits weighted with uncertainty. All error bars not shown are smaller than the size of the symbols.

Fig. 10
Fig. 10

Spectral power for various wavelengths as a function of propagation distance for amplified pulses with total incident power of 8.6 mW . Squares, circles, and triangles with apices up and down represent wavelengths of 700, 800, 810, and 820 nm , respectively. The solid lines are exponential fits weighted with uncertainty. All error bars not shown are smaller than the size of the symbols.

Fig. 11
Fig. 11

Absorption curve of water, as measured by Kou et al. [11] (lighter curve), and the deionized water sample, measured with the method outlined in Appendix A (darker curve). Also shown are absorption coefficients measured by laser oscillator (asterisks) and amplified pulses (circles) with total incident powers of 43 and 4.8 mW , respectively.

Fig. 12
Fig. 12

Total integrated power as a function of pulse chirp and duration as measured by the spectrometer for amplified pulses having total incident power of 8.6 mW . Pulse duration is 800 fs at 10 , 000 fs 2 , 35 fs at 0 fs 2 , and 400 fs at 5000 fs 2 . Where error bars do not show, they are smaller than the size of the symbol.

Fig. 13
Fig. 13

Spectral power for various wavelengths as a function of pulse chirp and duration as measured by the spectrometer for pulses with total incident power of 8.6 mW . Squares, circles, and triangles with apices up and down represent wavelengths of 700, 800, 810, and 820 nm , respectively. Pulse duration is 800 fs at 10 , 000 fs 2 , 35 fs at 0 fs 2 , and 400 fs at 5000 fs 2 .

Fig. 14
Fig. 14

Pulse spectrum shown as a compilation of narrower spectra generated by a mechanical slit in the Fourier plane of the pulse shaper. The dashed curve represents the shape of the spectrum when the mechanical slit is wide open; the curves, each about 5 nm wide, are generated by transversely shifting the mechanical slit in the Fourier plane of the pulse shaper.

Fig. 15
Fig. 15

Measured absorption curves for various water samples shown with the previously published data of Kou et al. [11], Segelstein [13], and Pope and Fry [10]. The uncertainty in the measured absorption curves is taken to be the spread in values in the immediate vicinity of a single point, which is ± 0.1 m 1 .

Equations (3)

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I ( ω , z ) = I 0 ( ω ) exp [ α ( ω ) z ] ,
T ( z ) = I 0 ( ω ) exp [ α ( ω ) z ] d ω ,
α ¯ = I 0 ( ω ) α ( ω ) exp [ α ( ω ) z ] d ω I 0 ( ω ) exp [ α ( ω ) z ] d ω .

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