DOI 10.1007/s10946-015-9489-9

Journal of Russian Laser Research, Volume 36, Number 2, March, 2015

SPECTROSCOPY OF OPTICALLY DENSE MEDIA BY FAST-TUNING LASER EXCITATION S. N. Andreev,1 A. V. Mikhailov,2 V. N. Ochkin,1,3 N. V. Pestovskii,1,3 and S. Yu. Savinov1,3 ∗ 1 Lebedev

Physical Institute, Russian Academy of Sciences Leninskii Prospect 53, Moscow 119991, Russia 2 Saint

Petersburg State University Universitetskaya nab. 7–9, St. Petersburg 199034, Russia 3 Moscow

Institute of Physics and Technology (State University) Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia ∗ Corresponding

author e-mail:

savinov @ lebedev.ru

Abstract We consider the possibility of using spectroscopy for the diagnostics of optically dense media by recording the time-dependent intensity of transmitted light under the action of a laser field with fastchanging frequency. We compare advantages of this approach with the traditional technique based on the Beer–Bouguer–Lambert law. Reasonably accurate extinction measurements of the order of k(ω0 )l ≈ 102 and greater are realizable with the help of tunable diode lasers.

Keywords: rapid passage of the absorption line, dense media, nonstationary effects, Beer–Bouguer– Lambert law.

1.

Introduction

Characteristic oscillations of the intensity of RF radiation after passing through an absorbing medium subjected to an external magnetic field with fast strength variation had been studied in the 1940s [1] and have found numerous successful applications in NMR medical diagnostics, nondestructive testing of materials, etc. Experiments with the fast shift of Zeeman or Stark absorption lines of media irradiated by a highpower laser at fixed frequency were undertaken in the 1970s in the IR optical region [2,3] and demonstrated similar effects. Accordingly, from the results obtained, it is possible to conclude that oscillations in RF and optical regions are of different nature. In the 1980s, optical experiments were rearranged and oscillation effects started to be observed not by the change of an external field but with the help of fast frequency tunable conventional diode lasers [4]. An appropriate theory was constructed. Later, in the 2000s, numerous experimental studies were stimulated by the invention of tunable quantum cascade lasers (see, e.g., [5–10]). As it looks now, the main applied interest of the above laser studies was to take into account the nonstationary effects as a disturbing factor in the precise high-resolution laser spectroscopy technique [9]. In this paper, we discuss the opposite – their active application for the spectroscopy of optically highly dense media. Translated from manuscript submitted on November 11, 2014. c 2015 Springer Science+Business Media New York 1071-2836/15/3602-0162  162

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Statement of the Problem

The results of measurement of the absorption coefficient k for a weakly transmitting medium are critically influenced by the finite experimental accuracy of light intensity determination [11]. This is qualitatively clearly seen in Fig. 1, where a set of spectral dependences of relative transmitted light intensity I/I0 for different optical densities k(ω)l is presented along with the normalized Doppler profile — the Beer–Bouguer–Lambert (BBL) law; it reads     ω0 + Δω I(Δω) l , = exp −k I0 ΔωD

(1)

where I0 = I0 (ω0 + Δω) is the radiation intensity of the laser source (it is assumed that the intensity does not depend on the frequency in the range of the absorption spectrum width), l is the length of the medium, k(ω0 )l = k0 l is the optical density in the center of the absorption line, and ΔωD is the Doppler width of the spectral line. These dependences can be registered by slowly changing the radiation frequency of the light source. At high k0 l ≥ 5, the dependences are saturated near the center of the line and at k0 l ≥ 50 even in the far line wings, so it is hard to distinguish them. It is possible to quantify the differences in the transmission curves for these dependences by introducing a parameter R   Ii (k0 l) Ii [(k0 l)∗ = 100] 2 1  ∗  − , R(k0 l, (k0 l) ) = √ I0 I0 n n

(2)

which is the normalized root-mean-square deviation of transmission I(ω, k0 l)/I0 . Here, (k0 l)∗ refers to the curve with the known value of k0 l, and n is the number of samples. In the example represented by curve 1 in Fig. 2, we use (k0 l)∗ = 100 and n = 281. We see that for k0 l from 10 to 40 the value of R exceeds 0.1 (10%). At the same time, in the vicinity of (k0 l)∗ = 100, these values are very small; for k0 l = 95 and k0 l = 105, they are R = 6.7 · 10−3 and R = 6.4 · 10−3 , respectively. In practice, to determine k0 l, it is necessary to compare experimental curves with calculated values according to the BBL relation (1) and then to minimize the value of R(k0 lexp , k0 lcalc ) (2). Thus, for a 5% accuracy of k0 l determination for k0 l ∼ 100, it is necessary to provide the accuracy of the intensity measurement ΔI/I0 ≈ 0.2% (it corresponds to 1/3R), which is a rather difficult task.

3.

Proposal

We apply the semiclassical description for the propagation of frequency-tunable light in an absorbing medium [4], where the radiation field is treated classically and the medium is considered as a number of noninteracting two-level quantum oscillators with bulk density N . It is assumed that the light beam is collinear with linear polarization, and the moments of transition in quantum systems are oriented in space in the direction of polarization of the incident radiation. The method of calculation of the electric field is based on the simultaneous solution of the classical wave equation and the von Neumann equation for a two-level system. In [4], the following expression was obtained for the complex amplitude of the

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Fig. 1. Spectral dependences of relative transmitted light intensity I/I0 for different optical densities k0 l (figures near the curves) along with the normalized Doppler profile.

Fig. 2. The dependence of the normalized rootmean-square deviation R(k0 l) on the optical density k0 l at (k0 l)∗ = 100. The classical BBL-based method (curve 1) and the method using fast-changing light frequency (curve 2).

electric field E(t, z) at the exit of the cell of length z: ¯ z) = 1 E(t, 2π

∞

∞ dν

−∞

−∞

  ¯  , z = 0) exp i ν(t − t ) − z [ν − iA(ν)c] , dt E(t c 2πk |d10 |2 γ A(ν) = 

∞ −∞

(3) dΔ g(Δ ) , −1 τ − i(Δ − ν)

¯  , z = 0) is the complex amplitude at the entrance of the absorbing medium, k is the absolute where E(t value of the wave vector of exciting radiation, Δ is a frequency detuning function also accounting for the inhomogeneous line broadening part, γ = N (ρ(t)00 − ρ(t)11 ) = γ0 is the population difference of states in coherent superposition, ρ(t)00 and ρ(t)11 are the diagonal elements of the density matrix, and τ is transversal relaxation time. Note that the described phenomenon does not relate to the redistribution of the population of the levels. We consider the case where the intensity of the laser field is too small to change the population during the action time — the characteristic time of interaction tsc = Δω/μ (μ is h the tuning rate) is much smaller than the Rabi nutation period TR = [12], where E0 is the electric E0 d10 field strength and d10 is the matrix element of dipole moment of transition |0 → |1. The radiation intensity can be easily calculated knowing the value of the complex amplitude of the electric field from (3); it is 1¯ ¯ ∗ (t, z). z)E (4) I ∼ E 2 = E(t, 2 For the radiation with linear frequency modulation, ω = Ω + μt,

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−T /2 ≤ t ≤ T /2,

T τ,

Δω = μt,

Δω/Ω 1.

(5)

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Journal of Russian Laser Research

For the inhomogeneous Doppler broadening, the frequency detuning function reads   2    2 ln 2 − Δ Δ . g(Δ, Δ ) = exp −4 ln 2 Δωd π Δωd

(6)

The optical density in the line center is described by the relation 4πkd210 k0 z = γz h

∞

−∞

τ −1 g(Δ = 0, Δ )dΔ . τ −2 + Δ2

(7)

The radiation intensity I should be calculated for time t = t − z/c, where z = l is the medium length. We assume that the radiation intensity is equal to unity in the absence of absorption. Figure 3 shows the calculated intensities of passed radiation in the case of inhomogeneous broadening of the absorption lines, where √ Δωl ln 2 = 0.1, Δωl = 2τ −1 . (8) Δωd The dimensionless velocity of the scanning rate μτV2 = 3.21 (the dimensionless scanning rate μτV2 is the ratio of the decoherence time τV to the scanning time tsc ) and the optical density in the line center k0 l = 95, 100, and 105. The time scale is normalized to the decoherence time τV = Δω −1 . It corresponds to the case of the CO2 absorption Doppler line at T = 300 K of the fundamental absorption band near 4.3 μm. The frequency tuning rate is 1.1 · 107 cm−1 · s−1 . We see that, despite the large optical density, radiation easily passes through the medium. In some phases of oscillations, the intensity even exceeds that of the radiation source. The physical cause of this is related to the induced Fig. 3. Calculated temporal dependences of intensities self-radiation of the medium and will be discussed of passed radiation with the parameters of calculations: elsewhere. Here, we would only mention that the μτV2 = 3.21 and k0 l = 95 (dotted curve), 100 (solid temporal structure of oscillations is rather sensitive curve), and 105 (dashed curve). to the value of absorption k(ω0 )l, if the other parameters of the medium and the frequency tuning rate are fixed. The normalized root-mean-square deviation R was calculated in the vicinity of (k0 l)∗ = 100 and is shown by curve 2 in Fig. 2. For k0 l = 93, 95, 97, 100, 103, and 105, the corresponding values of R are 0.270, 0.193, 0.15, 0.11, and 0.177. So, in order to determine k0 l, it is necessary, as in the above discussed case of BBL-based measurements, to compare the experimental curves with the calculated time dependences for the intensities of passed radiation, minimizing the value of R(k0 lexp , k0 lcalc ). In the latter case, however, to obtain the same 5% accuracy of determining k0 l for values of ∼ 100, it is necessary to provide the accuracy of measurement of the intensity ΔI/I0 ≈ 6%(1/3R). This can be easily done in modern experiments with tunable semiconductor lasers. Thus, we propose to measure the absorption of optically dense media not by recording the transmission of stationary media with the help of the BBL law but by recording the unique “fingerprints” of oscillations for the time-dependent intensity in transmitted light with fast frequency tuning.

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Conclusions

Absorption coefficient measurements in optically dense media with the help of the classical BBL law are based on a comparison of the input and transmitted light intensities at a fixed frequency and encounter problems. As an alternative, we consider the possibility of comparing the “fingerprints” in the time-dependent intensity of transmitted light with fast-changing frequency. The intensities to be measured are compared with the intensity of a transmitting light source and thus, can be determined with a reasonable accuracy close to that of the optical density k0 l restoration.

Acknowledgments The authors are grateful to Dr. A. A. Petrov and Dr. I. Yu. Tolstikhina (from the Lebedev Physical Institute of the Russian Academy of Sciences) for important discussions on the topic of this work. This work was financially supported by the Russian Science Foundation under Grant No. 14-22-00273.

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