We observe non-exponential absorption as a function of path length for pulses propagating in pure water. Two types of pulses with differing time duration, bandwidth, and repetition rate were compared with simulated absorption predictions. Deviations from exponential behaviour occurred when the launched pulse had a temporal width of 60fs and a repetition rate of 1 kHz. Under these conditions we observe more than 2 orders of magnitude less absorption after propagation through 6 m of water compared to Beer’s law prediction. No significant deviation was observed for launched pulses of varying bandwidth with temporal widths of 900fs and repetition rates of 80MHz.
© 2006 Optical Society of America
The conditions under which less than exponential absorption can arise from linear interactions in a dielectric has been analyzed by Crisp and Walmsley [1, 2]. Crisp’s discussion on deviations from Beer’s law was based on a study of 0π-pulse propagation through dielectric material. The predicted behavior of the propagating pulse suggests that energy from the main pulse evolves into new pulses with changing spectral and temporal properties over the path of propagation. These 0π-pulses have been associated with precursors, as discussed in . Precursor generation has been attributed to both ultra-fast pulse/material interactions  and broad bandwidths of the incident pulses . A signature of optical precursor formation is non-exponential absorption as a function of path length in a purely linear system. Crisp’s analysis used a two-level system whereas Walmsley used a more general non-stationary multilevel system. In both formulations the relationship between the temporal pulse length and the media’s dielectric relaxation was critical in determining if deviation from Beer’s law would occur.
In this work, we investigate the absorption with distance in pure water for pulses with different temporal pulse widths, bandwidths and repetition rates. Additionally, we compare our measurements to simulations that properly account for broadband Beer’s law behaviour as outlined in . We demonstrate that transient pulse formation is a function of the repetition rate in addition to the pulse width. Our observations indicate that the water acts as a time-variant linear system under excitation of femtosecond low power pulses.
To measure the absorption through pure water, a reflecting chamber was constructed to contain the water and allow varying path lengths by adjusting the entry angle of the beam. This chamber was based on the White cell spectroscopy concept of measuring long path lengths through small volumes of medium. The chamber is constructed of 316 grade stainless steel and the end mirrors were fabricated by evaporating chromium and silver onto a quartz glass substrate. In Fig. 1, only the coating protecting the mirror is visible on the front surface. The beam path length is controlled by varying the angle of entry through the uncoated section of glass on the front face. The mirrors were thoroughly characterized to account for reflection losses due to multiple passes through the chamber and characterized for losses due to varying reflection angles. The reflected signal was examined spectrally to verify multiple reflections did not impact the spectrum.
Two laser systems are used, the first system is a Spectra-Physics MaiTai diode-pumped, mode-locked Ti:S laser. This laser is tunable over a wavelength range of 730nm to 840nm and has a repetition rate of 80MHz. The second laser system used is a Spectra-Physics Tsunami laser system consisting of a diode-pumped, mode-locked Ti:S laser followed by a regenerative amplifier and parametric amplifier for wavelength tunability. This system is capable of producing 40fs pulses at a wavelength of 800 nm with a repetition rate of 1 kHz and an average power of 500mW. Measurements to determine the conditions resulting in non-exponential absorption included varying spectral and temporal widths of the pulses generated by these two laser systems.
The non-exponential absorption observed by Choi and Österberg was attributed to the broad bandwidth of pulses generated using a supercontinuum fiber . To verify whether or not this observation was due to broad spectral content, we coupled the output beam of the MaiTai laser into 30cm of Newport F-SV optical fiber, yielding a 50nm output spectrum with a temporal width of approximately 900 fs. For some of the measurements this bandwidth was reduced to 20nm using a bandpass filter.
Figure 2 diagrams the absorption measurement test setup with the MaiTai laser. Following a Faraday rotator, the beam is coupled into the optical fiber. The fiber output is collimated with a focusing lens and expanded using a 3x beam expander. Band pass filters are used to shape the spectrum before entering the water. The rejection ratio for the filters, made by Chroma Inc., is 10-5. The path length inside the pure water-filled chamber is varied by adjusting the entry angle of the beam. Absorption during transmission is determined by measuring beam energy before entering the chamber with a photomultiplier, then again after exiting the chamber. Considering negligible scattering in pure water (compared to absorption at these wavelengths), this method enables us to determine absorption as a function of path length.
The second test setup using the Tsunami laser system has approximately the same spectral properties, but has shorter pulses and a repetition rate much lower than the relaxation times associated with water. No spectral broadening optical fiber is used in this test set although band pass filters are used to shape the input spectrum. The beam is collimated and coupled into the reflecting chamber similar to the first setup.
Numerous characterizations were made to ensure that all interactions between the pulse and water were linear to verify that the non-exponential absorption observed is not the effect of any intensity dependent nonlinearities or detector limitations. The following control measurements were made; influence of two-photon absorption (TPA), stimulated Raman scattering (SRS) and self-phase modulation (SPM) as a function of power, threshold for thermal blooming, photomultiplier linearity and saturation, beam diameter in water and on detector, and neutral density filter linearity and saturation.
Intensity of the measured pulses were characterized to ensure that all measurements were performed below the threshold for two-photon absorption by measuring output intensity through a volume of water and plotting it as a function of input intensity. Performing this experiment using the MaiTai laser system, the curve remained linear over all intensities used in absorption measurements. When this characterization was performed with the Tsunami laser system, despite the high peak power, the curve remained linear to the maximum output of the laser. The spectrum of the two laser systems was constantly monitored to ensure no occurrence of SRS or SPM.
Thermal effects due to laser light heating the water as it propagates through could also contribute to nonlinearities. Thermal blooming, the effect in which heating of the water changes the refractive index of the medium, is an intensity dependent nonlinearity. Blooming was observed (by expanding the beam with a lens onto the wall) using the MaiTai laser when the 3x beam expander was not in place. No thermal blooming was observed when using the Tsunami laser system in part due to the ultra-fast pulse and slow repetition rate.
A Hamamatsu R958 photomultiplier (PMT) was used routinely for the transmissive absorption measurement. This detector was thoroughly characterized to ensure all measurements were taken within its linear region. This characterization required selecting a maximum operating intensity, not to be exceeded during any measurement using the PMT, and plotting output current as a function of PMT gain. The optimal gain set point was selected as the highest control voltage within the linear regime. Secondly, the linearity of the PMT/Keithley 485 picoammeter system was characterized. At low current values, the Keithley ammeter proved to have a nonlinear response that required correction.
The beam diameter was characterized to ensure that all the light coupled into the reflecting chamber and onto the PMT active area. Using the knife-edge method following Khosrofian and Garetz , the 1/e beam diameter was found to be 5.2mm for the MaiTai laser and 12.3mm for the Tsunami laser before the focusing lens. The PMT has a 4mmx20mm active area, large in comparison to the focused beam size, which was determined using the knife-edge method and lens focal length to be approximately 9μm for both the MaiTai and Tsunami lasers, respectively. Alignment of the focused beam spot onto the active area was critical for measurement consistency. Using a vertically adjustable stage, the PMT was mounted behind two pin-hole diaphragms and the focusing lens. Alignment through the two pin-hole diaphragms before the focusing lens ensured the beam spot was detected at the same point on the active area each measurement.
Final equipment characterization included measuring the accuracy and linearity of any neutral density filter used in the set up. No saturation effects or spectral variations were observed.
Beer’s law predicts that energy of a monochromatic wave will be reduced exponentially as a function of path length. To measure a deviation in absorption, the experimental data must be compared to an absorption function representing all of the spectral content present, rather than using a monochromatic approximation . Figure 3 shows input spectra for the MaiTai (left) and Tsunami laser systems (right), using 20nm bandpass filters for both lasers. Superimposed on the plots are the Segelstein absorption coefficient values as a function of wavelength . Within the wavelength region associated with the laser pulse, the absorption coefficient has a minimum at 800nm, marking the value to which the predicted absorption slope should converge after long path lengths for the case of exponential absorption. Temporally the MaiTai pulse width is 900fs and the Tsunami pulse width is 60fs (broadened from its original 40fs by the optics between laser and experiment). The two spectra have similar bandwidths, but differ in chirp and repetition rate.
Figure 4 shows the measured absorption function for pulses with spectral content shown in Fig. 3. The solid line labeled Exp.Decay is the predicted attenuation with distance for a broadband pulse when consideration is given to the fact that each wavelength within the pulse attenuates exponentially with with a different value of the exponent . The dotted reference line represents Beers’s law for the monochromatic decay at a wavelength of 800nm. The 900fs MaiTai laser pulses with a 80MHz repetition rate track the monochromatic decay for distances up to almost 6m. However, the 60fs Tsunami laser system pulses, with a lower repetition rate, clearly deviate from the longer pulse behaviour and the predicted absorption function, with more than 2 orders of magnitude difference at a distance of 5.7m.
The absorption measurement using the full 50nm spectrum from the MaiTai pulses propagating through the fiber produced similar result to the 20nm MaiTai result shown in Fig.4.
4. Discussion and Conclusion
The key observation from our measurements is that two different laser systems, both operating in a linear regime, centered at the same carrier wavelength, with the same spectral properties but different pulse widths and repetition rates produce drastically different absorption behaviour. The increase of bandwidth from 20nm to 50nm for a 900fs long pulse with a 80MHz repetition rate does not significantly alter the absorption properties. However, a 20nm, 60fs pulse at a repetition rate of 1kHz has more than two-orders of magnitude more remaining energy after propagating 5.7m through pure water than the longer, high repetition rate pulses.
The dependence of the absorption with distance on the temporal pulsewidth and repetition rate is likely explained by the different relaxation processes in water. The dielectric properties of water in the visible and near-IR region of the spectrum are largely controlled by hydrogen-bond interactions. The time scales for intra- and inter-molecular motions range from 60fs  for librational motions up to 740fs  for vibrational stretching modes for overtones of excited OH bonds. At the same time there are larger collective modes which regulate relaxation processes in excess of 70ns in water . In view of these relaxation times we suggest that the non-exponential absorption with distance is due to the 60fs pulse being to short for the vibrational transitions to occur, and the slow repetition rate is necessary for the water to regain equilibrium before the next pulse arrives.
It is clear that more measurements are needed where pulse width and repetition rates are altered, however, based on our observations in this paper, temporal properties, rather than broad spectral content dominate the absorptive properties of water.
We thank T.Davis for technical assistance and Profs. E.Garmire and U.Gibson for helpful discussions.
This work was partially supported by NIST, grant 60NANB4D1142.
References and links
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