# Peculiarities of Propagation of Ultrashort Laser Pulses and Their Use in Atmospheric Sensing

## Lidar Control in Problems of Atmospheric Sensing by Ultrashort Pulses

Permanent monitoring of the state of the atmosphere, as well as technogenic pollutions, dangerous types of bioaerosol, which are indicators of chemical and biological contamination, as was mentioned in Chapter 1, is one of the first-priority problems in atmospheric optics because information of this kind is necessary for the prediction and improvement of the environmental situation both in particular regions and on the planetary scale. The existing global tendencies imply continuous development of remote sensing methods as the most efficient technology for monitoring of the atmospheric state with the use of ground-based, airborne, and spaceborne stations.

Methods of lidar sensing occupy a particular place in remote sensing. The understanding of the fact that the radiation with wavelengths different from the wavelength of laser radiation carries information about the composition of matter in the target region was very important for application of lasers in remote sensing. Such specific features of lasers, as high power, monochromaticity, short pulse duration, and high directivity of an optical beam, have been tested, in the first turn, in atmospheric sensing. Results of laser sensing of the atmosphere have shown that lidar systems can be used to detect and measure parameters of atmospheric constituents of both natural and anthropogenic origin [1].

At the end of the twentieth century, large-scale investigations of high-power femtosecond laser pulses with aerosol-gas constituents of the atmosphere were started to provide for the long-distance propagation of optical radiation and to develop new methods for sensing of gas and aerosol characteristics [2—5]. Most interesting phenomena arise at the propagation of radiation with femtosecond pulse duration and power exceeding some critical level. In this case, the spectral, temporal, and spatial characteristics of laser radiation change significantly, accompanying the formation of light and plasma channels—filaments, in which the intensity of laser radiation is concentrated up to the level of optical breakdown in air (Th0^{14} W/ cm^{2}). The phenomenon of filamentation is accompanied by generation of supercontinuum radiation, that is, broadband radiation extending over the range from 0.3 to 4 pm in the wavelength scale. A pulse of this “white” radiation is considered as a promising source for laser sensing and environmental monitoring.

Physical methods for detection, identification, and quantitative estimation of various gas and aerosol constituents of the atmosphere and atmospheric parameters with the use of existing optical effects, in particular, laser-induced fluorescence, spontaneous and stimulated Raman scattering, Doppler shift, spectral absorption, anisotropic scattering at atmospheric inhomogeneities, and effect of appearance of conical emission of broadband radiation at the propagation of femtosecond pulses are continuously improved.

The effect of an extremely high-power femtosecond optical pulse on aerosol particles was studied under laboratory conditions. The scattering matrix and the scattering phase function of an aerosol particle in an extremely high-power optical field were studied experimentally. Field studies of physical characteristics and the structure of high-power femtosecond laser radiation at open-air paths till several kilometers long were carried out [6].

The Teramobile international research group [5] was the first to conduct the atmospheric sensing by femtosecond pulses up to heights of 10—15 km. The new field—femtosecond atmospheric optics—has been actively developed. Within the framework of this research field, physical principles of remote monitoring of microphysical and optical characteristics of atmospheric aerosol by femtosecond pulses with the use of intra-atmospheric broadband source of radiation (supercontinuum) are actively developed.

During the recent decades, this field was also intensively developed in Russia, which has a wide experience in atmospheric sensing and developed methods for solution of inverse atmospheric-optical problems, in particular, in multifrequency sensing [7—9]. Methods for control of the filament structure of radiation with the aid of spatial and temporal focusing of a laser beam, as well as with the aid of systems generating phase-modulated pulses, were tested [10]. In References 11—13, one can find a review of the state of the art in investigation of the phenomena of filamentation of high-power femtosecond laser radiation in transparent media. In addition, these works study the mechanisms of supercontinuum generation and spatial distribution of the sources of supercontinuum radiation during the propagation of a high-power femtosecond laser pulse in liquid and gas.

In Reference 14, a model of coherent scattering of high-power laser radiation at an ensemble of water aerosol particles oriented at problems of femtosecond nonlinear optics was considered. It is shown that the appearance of maxima in the intensity distribution of a laser beam at coherent scattering of radiation at aerosol particles can lead to generation of a random set of filaments.

In References 15—18, the additional effect of aerosol particles on an increase of the local field in the incident radiation was considered, thresholds of optical breakdown of a transparent particle were determined, and effects of the interaction of laser pulses with liquid-droplet aerosol were demonstrated.

In the series of works (see References 19—23), the problems of propagation of a high-power laser pulse in the atmosphere were considered with allowance for the multiple scattering, and the efficiency of using the white-light lidars for sensing of the molecular atmosphere and microphysical parameters of cloudiness was assessed.

The use of the effect of filamentation with formation of an intra-atmospheric broadband source of pulsed sensing radiation (supercontinuum) turns out to be the most attractive for the solution of problems of remote sensing. Approaches for the use of this kind of radiation in the laser sensing of the atmosphere are developed to estimate the characteristics of molecular and aerosol components. Problems of atmospheric sensing by femtosecond pulses require revision of the approach for the analysis of echo signals: it is necessary to invoke practically all known sensing methods both active and passive and, correspondingly, formulate new sensing equations.

The form of the lidar equation depends on the type of laser radiation interaction with the atmospheric medium. The used form of the lidar equation is that applicable to measurements with high spatial resolution [1,6].

As is known, if the sensing is carried out by pulses in an ordinary mode with the intensity level lower than the critical one and the pulse duration exceeding the level at which it is necessary to take into account the dependence of optical characteristics of the atmosphere on the pulse duration, then it is possible to solve the lidar equation in the known forms, namely, scattering, differential absorption, and fluorescence.

As an ultrashort radiation pulse having the high energy density propagates in the atmosphere, effects of nonlinear optics manifest themselves. The nonlinearity is most pronounced at the laser wavelength and at wavelengths of neighboring spectral ranges. Spatially, the nonlinearity is pronounced at the initial part of the path. High intensities in the filamentation zone turn on the processes of multiphoton absorption and phase self-modulation, which leads to spectral broadening, that is, supercontinuum generation. A part of the pulse energy is distributed over a wide, consisting of several spectral octaves, range, but to the edges of the distribution the energy decreases sharply, by several orders of magnitude.

The long-wavelength (i.e., Stokes) part of the spectrum is localized near the filament axis and takes part in the process of self-focusing, while the short-wavelength (i.e., anti-Stokes) component (generation of conical emission) is partially localized in the axial zone and partially experiences divergence with the linear wavelength dependence of the angle. The spectral components are coherent with each other. In the short-wavelength part, a ring structure is observed and the interference interaction of spectral components of different filaments is pronounced. The pulse continues to propagate in the direction of sensing. With allowance for the broadening and energy loss with time, the supercontinuum generation continues until the pulse intensity becomes lower than some critical value.

The spectral composition of the radiation scattered toward an observer is determined by the spectrum of the initial pulse and the composition of the propagation medium. The possibility of analyzing the absorption of broadband supercontinuum radiation develops the idea of a multicomponent laser. The broadband feature of the radiation of an intra-atmospheric source assumes the possibility of applying the known and specifically developed laser sensing methods to the determination of qualitative and quantitative composition of the atmosphere.

The spectrum recorded by a femtosecond lidar can be conditionally divided into the short-wavelength part (laser spectrum) and the long-wavelength part. The spectrum of the backscattered lidar signal in the laser wavelength range contains the scattered radiation of the femtosecond pulse and an addition due to supercontinuum. The short-wavelength and long-wavelength parts contain the supercontinuum spectrum. The lidar equation in its traditional form should be complemented with an additional term responsible for radiation of the intra-atmospheric source.

From the viewpoint of experimental geometry, the path of sensing by extremely short pulses can be conditionally divided into parts by the “linear-nonlinear” attribute. These parts are localized in length: the field of view of the recording system can include the section from a radiator to the filamentation zone, filamentation and generation of conical emission, the section illuminated by supercontinuum radiation, and the linear sensing pulse with possible partial overlapping. At these sections, the probability of illumination by the supercontinuum radiation varies from 0 to 1. At the initial section, the probability that the lidar system receives the supercontinuum radiation depends on the geometric form-factor and the ratio of the pulse intensity and the critical intensity in the conditional filamentation channel. This probability should be close to unity at all points of the path located farther than the point of super continuum generation. In the presence of several filamentation channels, the energy fractions distributed over the sensing pulse spectrum are summed. It is important to point out that the component at the laser wavelength is presented in the backscattered signal from all parts of the path: before the filamentation zone, and in the filamentation zone—in the nonlinear sensing mode, from farther parts—the rather powerful sensing pulse in the linear mode. It is necessary to take into account that the transition to the linear mode is caused not only by the energy loss for filamentation, but also by the temporal

broadening of the initial pulse. After the boundary of the zone of conical emission generation, the supercontinuum radiation is presented in the backscattered signal with the unit probability as an additive component to the attenuated radiation at the laser wavelength.

The experience of the Teramobile international research group shows that the most significant results are obtained at the combination of the powerful femtosecond system and the powerful astronomical telescope. For example, in Reference 24, the telescope with 2-m mirror was used as a receiving system in the bistatic lidar scheme with a separation of 30 m between the transmitter and receiver.

For the linear case, the laser radiation power scattered toward an observer (subscript L) recorded as the pulse with duration t_{l} and initial energy *E _{0}* passes the distance

*R*can be presented in the traditional form:

where

*k(R)* is the vertical profile of the sum of extinction coefficients in the atmosphere for the laser and detectable wavelengths, the two-side (two-pass) extinction coefficient, *k(R) = k(X _{L}, R) + k(X,* R);

t_{l} is the laser pulse duration;

- ?(R) is the geometrically justified probability that the radiation from an object located at the distance
*R*reaches the detector (geometric form-factor); - ?(A) is the spectral transmission coefficient of the optical system;

A_{0} is the effective area of the optical receiving system (account for the form-factor);

*(3**(*A_{l}*, A, R)* is the volume backscattering coefficient;

*P*l *= E _{l}/*t

_{l}is the power of laser radiation.

For the case of elastic (Mie or Rayleigh) scattering, the observation wavelength coincides with the laser wavelength. Just the linear version of the lidar equation is used in Reference 25 with the only difference that the equation is written for the photon number, and the supercontinuum formation point is introduced to the integration limits in determination of the extinction coefficient. This form is used in analysis of the results of sensing of path points lying above the generation point by the supercontinuum radiation with ignoring of nonlinear effects. The detection time achieves the values from 1.5 to 15 ps. The overlapping of the field of view of the receiving telescope (i.e., a mirror of 40 cm, frequency *—f/3,* bistatic lidar scheme, focusing to the end of a waveguide 1 mm in diameter) and the divergence angle of the supercontinuum radiation was provided by the geometry of the experiment, and thus the probability of overlapping of these angles was believed to be equal to unity.

The way to take into account nonlinear effects in the lidar equation that was proposed in Reference 26, in which the emphasis was at the transformation of the backscattering component for consideration of nonlinear effects in determination of aerosol particle size spectra. Peculiarities of propagation, such as self-focusing, were not considered. Consequently, the extinction component remained in its original form, which is unreal from the physical point of view because of the high power of radiation.

The lidar equation for sensing of supercontinuum radiation (SCR) was considered in Reference 27. Particular attention was paid to consideration of the influence of the function of overlapping of the solid angles of field of view of the recording and the receiving instrumentation on the decrease of signal power under conditions of SCR self-focusing and self-channeling in the atmosphere. High intensities achieved in filaments also lead to nonlinear extinction of pulses. With allowance made for these conditions, the modified equation was presented, which takes into account the influence of the form of the sensing beam and the extinction due to multiphoton ionization on nonlinear extinction and backscattering components. The assumption of the low influence of pulse energy loss on supercontinuum generation is used, and the strong narrowing of the spectrum on the both sides of the main wavelength is taken into account.

With allowance made for the spatial characteristics of the spectrally distributed signal, we come to the conclusion that the geometric form-factor of lidar (probability of overlapping of the field of view angle of the receiving system and the divergence angle of the sensing pulse) is significantly different for the short-wavelength and long-wavelength spectral ranges: the initial divergence of the collimated laser pulse can be equal to fractions of milliradian, while divergence in the short-wavelength supercontinuum spectral range is 0.1—0.2°. Consequently, it is necessary to take into account the dependence of the geometric form-factor on the distance and on the wavelength. In particular, this dependence can be minimized through introduction of at least two channels for receiving of the backscattered sensing signal to the lidar scheme. Each channel should have the possibility of recording of the spectrally distributed signal and the corresponding form-factors. The requirement of temporal resolution appears to be specific. As a rule, photomultiplier tubes with the temporal resolution (for detection) from fractions of microseconds to few microseconds are used as measurement converters. With allowance for the observation experience (Teramobile), the supercontinuum channel operates in the accumulation mode with times up to several tens and even hundreds of seconds. It is natural that the signal in this case is received from a chosen part of the path, which is implemented either through time gating in the receiver or in the bistatic scheme through fixation of the angle between optical axes of the receiver and the radiator. In any case, the problem of influence of multiple scattering remains nonsolved. With allowance for temporal restrictions at the reception of optical systems by measurement converters employing the photoeffect (fractions of nanosecond are the best samples), we come to the conclusion that modern lidars can provide for the effective reception of backscattered signals in the accumulation (time averaging) mode from path sections located farther than the filamentation zone.

Thus, it is possible to adjust the lidar so that the signals from the path part with predominance of nonlinear effects (intensities much higher or equal to the critical value) fall within the dead zone. The proper choice restricts the spectral range of reception and the form-factor value. Then, backscattered signals from the point of supercontinuum generation *R _{SC}* to the control point

*R > R*and in the backward direction—from the entire path to the receiver of radiation at the point

_{SC}*R =*0 will be assigned to the results of sensing. In general, the lidar equation can be reduced to the linear one, but the scattering signals will be influenced by nonlinear effects at the initial parts of the path. The paradoxicality of the situation is that the generation of supercontinuum radiation has the nonlinear character, while backscattered signals can be considered as linear signals with allowance for the spatial-temporal scales of a filament. In turn, at organization of the channel of parallel sensing by long pulses, it is possible to compare results and to draw conclusions about, for example, the relation of the extinction coefficients in the linear and nonlinear modes [28].

To estimate comparative characteristics of the femtosecond lidar, analogous to that described in experiments of the Teramobile group, and the traditional nanosecond lidar [29], the amplitude of the backscattered signal was calculated for the set of wavelengths from the spectral range of supercontinuum and the wavelength of sensing by the nanosecond lidar (Figure 5.1). The calculation was made with the use of reconstructed profiles of the extinction coefficient, the scattering and backscattering coefficient, and the lidar ratio. The calculation logics and algorithms correspond to the traditional linear approach, from which follows [30-32]:

.. .With an accuracy sufficient for practical needs, the process of laser beam propagation in the aerosol atmosphere can be described with only three optical characteristics available: the extinction coefficient, scattering coefficient, and scattering phase function.

For the nanosecond lidar, the following parameters are specified: pulse energy of 10 mJ, pulse duration of 10 ns, wavelength of 0.53 pm, and receiver’s aperture of 0.305 m. The range of 1000 m is determined for the practically cloudless atmosphere (visibility range >30 km) for the signal-to-noise ratio equal to 10. For the conditional femtosecond lidar, the following characteristics were chosen: wavelength of 0.8 pm, pulse energy of 10 mJ, pulse duration of 100 fs, pulse repetition frequency of 10 Hz, and receiver’s aperture of 1 m. Photodetector (conditional) is of mark FEU-83 PMT. The point of supercontinuum generation was taken at a distance of 0.1 km from the source, and the total coefficient of energy transfer into the energy of supercontinuum is 0.23 of the energy of femtosecond pulse at this point of the path. For the chosen supercontinuum wavelengths, the energy transfer

Figure 5.1 Vertical profiles of the ratio of the backscattered signal amplitude to the detector (FEU-83) noise—signal-to-noise ratio, current mode of operation. Designations: SN532—for the nanosecond lidar, wavelength of 532 nm (control version); SN500, SN610, SN670, SN1060—for supercontinuum wavelengths of 500, 610, 670, 1060 nm, respectively; *h* is distance in km. In the top part: solid line—femtosecond lidar, dots—nanosecond lidar. Horizontal dashed lines show the range of satisfactory values of the signal-to-noise ratio from 1.5 to 10 and conditional level of 1000, limiting the reception of signals from the near zone of the path.

coefficients were taken to be 0.01, 0.04, 0.08, and 0.1 for wavelengths of 0.5, 0.61, 0.67, and 1.06 pm, respectively [22].

Figure 5.1 shows vertical profiles of the signal-to-noise ratio, the results of model calculation of the amplitudes of backscattered signal from different points of 2-km path for the following conditions: meteorological visibility range of 5 km at the ground level (weather type: summer period, stable haze, visibility range of more than 4 km).

One can see from the figure that, in general, for the given initial conditions, the recording of spectral signals from supercontinuum is possible for path points from 0.8 km (for wavelength of 0.5 pm) to 1.6 km (for wavelengths of 0.67 and 1.06 pm). The range of the femtosecond lidar in the visible and near-IR range is comparable with the range of the traditional nanosecond lidar and determined by the energy transferred into the supercontinuum spectrum in the process of filamen- tation of the initial pulse and by the parameters of the receiving system.

In the generalized form, the lidar signal from the path points, from *R _{SC}* to

*R,*recorded by the lidar receiving system in the case of illumination by the supercontinuum radiation, can be represented as a sum of two components:

where *P _{X}(R, X) =* Pj(R) is the ordinary lidar equation at the wavelength of laser radiation X =

*X*

_{L}:

where *E _{x}* is the residual energy of the laser pulse after generation of supercontinuum,

*R*is the distance to the point of appearance of supercontinuum,

_{SC}*R > R*is the effective receiver area,

_{SC}, A*A =*?(R)A

_{0}, A

_{0}is the aperture of the receiving system,

*?*is the geometric form-factor of the lidar,

*c*is the speed of light,

*в*is the vertical profile of the volume backscattering coefficient,

*k*is the vertical profile of the extinction coefficient, and

*X*is laser wavelength. This equation allows ordinary procedures of inversion for the backscattering coefficient or extinction coefficient.

_{L}Keeping in mind the peculiarity of radiation generation in a wide spectral range—the generation starts at the remote (from the receiver) path point *R*_{SC}—the echo signal from supercontinuum radiation can be described as follows:

where

is the energy of supercontinuum radiation pulses in the wavelength range ДХ, *? _{SC}* is the spectral density of supercontinuum energy; ДХ is the band of wavelengths detected by the receiving system of the femtosecond lidar;

*k(x, X) = k(x, X*+

_{L})*k(x, X)*is the total extinction coefficient at the sensing wavelength and the detected wavelength.

Equation 5.4 describes the result of scattering of supercontinuum radiation in the atmosphere.

Let us analyze the possibility of inversion of Equation 5.4 for determination of the concentration of gas components. We use the approach of the well-known differential absorption and scattering (DAS) method, which is based on detection of ^{four si}g^{nals: p}*sc*^{(R} J; P_{SC}* ^{(}R + ^{AR}* AJ;

^{p}*sc*

*А*

^{(R}_{#});

^{p}*sc*

^{(R}+^{AR}*ff*

*).*

^{Her}e

_{o}*n* lies within the profile of the absorption line of the sought gas, *A** _{o}ff* lies in a wing or outside the absorption line; АА is chosen based on the widths of absorption bands of detected gases. For simplicity, we assume that the system is well aligned, that is, ?(R) = 1.

We can write the atmospheric extinction coefficient with the absorption by the analyzed gas separated in the explicit form *K(A) = a (A) +* a (A), where a (A) = *N -S* (A), *N* is the concentration, and S(A) is the absorption cross section per one molecule. Then we can formulate the following equations:

We introduce now the average values *a* and *N* in the range *AR* and solve this system of equations for *N*:

As a rule, *A _{on}* and

*A*are close and, from the viewpoint spectral dependences of

_{o}ff*в*and

*a,*we can believe that в(

_{п}) =

*@(.A*and

_{a}ff)*a(A*Then

_{an}) = a(Aff).

We have derived the equation analogous to that for ordinary lidar employing the DAS method. An advantage of the femtosecond lidar is that the spectrally wide supercontinuum radiation overlaps absorption bands of many gases, thus allowing the multigas analysis of the atmosphere. In Equation 5.8, there is no dependence on the spectral density of supercontinuum energy. This is valid only in the case when the signals *P _{SC}(R,* A

_{on}) and

*P*AR, A

_{SC}(R +_{on}), as well as

*P*and

_{SC}(R, Aff*P*AR,

_{SC}(R +*A*), are detected from the same generation pulse. The coefficient of energy conversion into supercontinuum radiation determines the potential of the white-light lidar.

_{o}ffIn many problems, it is needed to determine the profile of the extinction coefficient of the atmosphere related to the variability of the atmospheric aerosol composition [33]. In this case, it is possible to use the algorithms of mutifrequency analysis [34], which provide information about the microphysical properties of particles. The supercontinuum radiation is ideal for the application of these algorithms for some classes of atmospheric aerosol. Consider the solution of lidar Equation 5.4 for the extinction coefficient with the emphasis, for example, at the Klett method [35]. In Equation 5.4, one can pass to logarithmic and square-amplified return signals S(R, A) = ln [P_{K}(R, A) • R^{2}] and, in addition, S(R) is taken to be known at the limited range *R _{n}* (e.g., the value is close to the noise level). Then the difference

depends only on the atmosphere, and the solution can be written in the form

where *g* is the parameter of relation between the backscattering and extinction coefficients for the elastic scattering *(g* depends on the lidar wavelength and specific properties of scattering centers; it varies in the range 0.67—1.0). The value of *g* can also be estimated from the condition of stability of solutions [1].

Thus, we can note that the inversion of the lidar equation with supercontinuum does not impose any specific formal differences on the signal processing algorithms. However, to be noted are possible limitations of the presented equations attributed to the contribution from multiple scattering, which is proportional to the field-of- view angle of the lidar receiving system. The point is that the necessity of complete interception of the supercontinuum radiation cone (with an angle degree exceeding 0.1°) requires the corresponding values of the fields of view of the receiving system, which explains the schemes of most successful atmospheric experiments with the use of astronomical telescopes having the high light-gathering power and small field-of-view angles. The consideration of the influence of multiple scattering and backscattered signals from the zone of development of nonlinear effects on signals of the femtosecond lidar is a future task.