ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2015, Vol. 79, No. 12, pp. 1507–1512. © Allerton Press, Inc., 2015. Original Russian Text © D.Yu. Zagursky, I.G. Zakharova, V.A. Trofimov, 2015, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2015, Vol. 79, No. 12, pp. 1719–1724.

Propagation of FewCycle Pulses in Homogeneous and Nonhomogeneous Media D. Yu. Zagursky, I. G. Zakharova, and V. A. Trofimov Moscow State University, Moscow, 119992 Russia email: [email protected] Abstract—The interaction between a fewcycle pulse and an optically extended multilevel medium is ana lyzed via computer simulation. The cascade mechanism behind the excitation of the medium’s high energy levels is determined and the way in which it is affected by the pulse’s absolute phase is investigated. The effect structural disordering has on the absorption spectrum of a multilevel medium is also analyzed. DOI: 10.3103/S1062873815120254

INTRODUCTION It is well known that the interaction between fem tosecond (fewcycle) pulses and media is an important problem of modern laser physics [1–6] for at least two reasons. First, their short duration allows investiga tions of different ultrafast processes. Second, they can be of high intensity (their electric field can exceed their intraatomic field); for such pulses, the electric field of an atom is a perturbation. There are also other features that are specific only to such pulses. For example, the medium’s response depends on the abso lute phase of the fewcycle femtosecond pulse [2, 3]. If the pulse duration does not exceed some critical value, the medium’s response occurs at the characteristic fre quency of the medium even when the optical radiation frequency is quite different. In addition, hysteresic dependences of the most intense spectral component of the medium’s response on the amplitude of an inci dent pulse are observed [7]. The widely used Duffing model of a medium’s response then cannot be applied to a fewcycle pulse [8]. Finally, highfrequency sub pulses that might have a solitonic form are possible upon the propagation of a fewcycle optical pulse in a medium with saturating potential [9].

responsible for the interaction between fewcycle pulses in different frequency ranges (optical, tera hertz, and micron [15, 16]) and a substance is there fore an important problem. In this work, computer simulations of the propagation of a fewcycle pulse in a resonantly absorbing medium (especially a disor dered structure) are analyzed. Such problems are encountered when identifying substances by subject ing them to wideband terahertzrange radiation. Treatment is done using 1D Maxwell equations and the numerical algorithms proposed in [17–19]. FORMULATION OF THE PROBLEM A quasiclassical system of Maxwell–Bloch equa tions in dimensionless form was used to describe the interaction between a pulse and a medium:

∂H = − ∂E , ∂D = − ∂H , ∂t ∂z ∂t ∂z D = E + 4πP, ∂ρ mn + (γ mn + iωmn )ρ mn ∂t = iα E (d mnρ qn − ρ mqd qn ),

∑

The absolute phase of a fewcycle optical pulse can also substantially affect the generation of the third har monic in a medium with cubic nonlinearity [10, 11]. The response in this case can also occur at even fre quencies [11], which does not happen with pulses of longer duration.

(1) (2)

(3)

q

∂ρ mm + ∂t = iα E

∑ (W

mqρ mm

− W qmρ qq )

q

∑

(4)

(d mnρ qm − ρ mq d qm ),

q

Femtosecond optical pulses (including fewcycle pulses) display new features when interacting with induced periodic temporal structures (i.e, temporal photonic crystals [12]). New approaches are being developed for analyzing this interaction, especially when it is linear [13, 14]. Studying the mechanisms

P =

∑ (d

mnρ nm ).

(5)

mn

Here, E is the electric field strength; D is electric induction; H is the magnetic field strength; ρmn are density matrix elements; t is time; and z is the coordi

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nate along which the electromagnetic pulse propa gates. The substance lies in the layer 0 < z < lmedium. As is well known, diagonal matrix elements ρmm corre spond to the number of the substance’s molecules found in state m and are referred to as the population of level (N). Parameters γmn and Wmn are the coeffi cients of transverse and longitudinal relaxation; dmn is the matrix of the dipole moments of transitions; ωmn is the matrix of the frequencies of transitions between energy levels of atoms; and α is a dimensionless coef ficient that characterizes the interaction between the electromagnetic field and the substance. It depends on the physical parameters of the problem, 4π nd 02 (6) ,  ω0 where d0 is the dipole moment and ω0 is the character istic frequency;  is the Plank constant. The effect the medium has on the field is considered through polar ization P (Eq. (5)). At the initial moment in time, the Gaussian elec tromagnetic field pulse propagating to the right is α=

E(z, t = 0) = H (z, t = 0) ⎛ (z − z 0 )2 ⎞ (7) = A0 exp ⎜ − ⎟ cos(ω p(z − z 0 ) + ϕ0 ), 2 τp ⎠ ⎝ where A0 is amplitude; z0 is the initial position of the pulse, (z0 < 0) is its center, τp is its duration, ωp is its fre quency, and ϕ0 is the initial phase. The substance’s molecules at the initial moment in time remain in the ground state: (8) ρ11 = 1; ρ mm = 0 at m ≠ 1. Equations (1)–(8) describe the propagation of the of pulse from the vacuum left of the layer (z < 0) of the medium through layer (0 < z < lmedium) of the medium to the region of the vacuum to the right of the medium (z > lmedium). We are interested here in the behavior of the populations of the energy level in cross section z = lmedium/2 of the medium, and in the time dependence of the electromagnetic field’s strength to the right of the layer (z = lmedium + 1) of the medium. The spectrum of the electric field passing through the substance is compared to the spectrum of the incident electromag netic pulse. CASCADE MECHANISM OF ENERGY TRANSITIONS The shape and spectrum of a pulse change after passing through a medium. Gaps corresponding to the lines of absorption by the medium appear in the spec trum; if a fewcycle terahertz pulse passes through the substance, frequencies lying outside the initial spec trum appear in its spectrum. Spectral peaks caused by the radiative transitions of molecules from their high

energy levels to their ground states are observed in par ticular. These lie beyond the initial pulse spectrum, even though there are no spectral components in the incident pulse capable of causing a direct transition from low to high energy levels. We emphasize here that these energy levels of molecules can be excited via a cascade of successive energy transitions between neighboring energy levels whose frequencies lie within the initial spectrum of the pulse. It should be noted that there is virtually no multiphonon absorption in this case, due to the low intensity of the incident pulse. To illustrate the functioning of this mechanism of the excitation of highenergy levels, we conducted a computer simulation of the interaction between a few cycle terahertz pulse and a medium containing four energy levels. The medium was characterized by the parameters α = 67.54; dipole moments dmn = 0.02; coefficients of transverse relaxation γmn = 0.5; and coefficients of longitudinal relaxation Wmn = 10–7. The transition frequencies can be seen in Fig. 1b. Let us first consider a case in which the pulse spec trum overlaps all possible frequencies of energy transi tions in the medium (ωp = 10, E0 = 2; and τp = 0.5). The shape of the initial pulse is presented in Fig. 1a. The spectra of the pulses incident on the layer and passing through it are shown in Fig. 1b, where the ver tical dotted lines denote the transition frequencies in the medium. Absorption bands are clearly seen in this figure. The evolution of the population of energy levels over time is shown in Fig. 1c. Note that the popula tions of all four energy levels start to grow simulta neously when the pulse passes the observation point. If the pulse has a narrow spectrum (Figs. 2a, 2b) that does not overlap the transitions from the ground state to the third and fourth energy levels (ωp = 5, E0 = 1; and τp = 2), energy transitions 4–1 and 3–1 are still visible in the spectrum of the pulse transmitted through the layer (Fig. 2b). The evolution of the pop ulation of energy levels over time is shown in Fig. 2c, where we can see that the population of high energy levels grows, even though direct transitions are forbid den. Note that the populations of the levels begin to increase successively, so higher energy levels start to fill only after the lower levels are quite full. This figure clearly illustrates the operation of the cascade mecha nism of high energy level excitation. EFFECT OF THE ABSOLUTE PHASE OF AN INCIDENT PULSE The response of a medium to a fewcycle pulse depends on the absolute phase of the incident pulse. To identify a substance with a terahertz pulse, we must analyze the dynamics of the absorption of the incident pulse’s energy as a function of the its absolute phase. We performed a number of numerical calculations for

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PROPAGATION OF FEWCYCLE PULSES E(z) 2.0

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E(z) 1.0

(a)

(a)

1.5 0.5

1.0 0.5

0

0 –0.5

–0.2

–1.0 –1.5 –4 –2

0

2

4

6

8

(b)

E(ω) 1.0

–1.0 –15

10 z

–10

–5

0

5

10 z

(b) E(ω) 1.0

3–4 0.8

3–4 0.8

2–4

2–4

2–3

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1–4

0.6

1–3 0.4

1–2

2–3 1–4

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0.2 0.2 1–2 0 0

5

10 (c)

N(t) 1.0

15

ω

0

5

10 (c)

200

400

N(t) 1.0

15

ω

0.8 0.8 0.6 0.6 0.4

0.4

0.2

0

0.2 100

200

300 t

Fig. 1. (a) Shape of the fewcycle pulse incident on our medium; (b) spectra of incident (solid line) and transmit ted (dotted line) pulses; (c) evolution of the population of medium energy levels. The vertical dotted lines represent the transition frequencies.

0

600

800 t

Fig. 2. (a) Shape of the incident fewcycle pulse; (b) spec tra of our incident (solid line) and transmitted (dashed line) pulses; (c) evolution of the medium’s population of energy levels upon cascade excitation of its high energy levels. Vertical dotted lines represent the transition fre quencies.

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phases. It is easily seen that lower frequency lines (e.g., 1–6 and 1–7) dominate when ϕ0 = π/2.

(a) E(z) 1.0

DISTORTION OF THE PULSE SPECTRUM IN LAYERED STRUCTURES

0.5

Terahertz spectroscopy is important because the transparency of many substances in this range of fre quencies allows us to identify substances imbedded in other materials. However, even if the packing (e.g., paper) is transparent because of its macroscopic struc ture (layers, variations in density), it can behave as a disordered structure with respect to terahertz radia tion. We performed computer simulations in which a twolevel medium was placed between two structural layers with random dielectric permeabilities. For these layers, we assumed that

0 –0.5 –1.0 –12

–10

–8

–6

–4

–2 z

(b)

E(ω) 1.0

D(z, t ) = ε(z )E (z, t ),

2–7 2–6

6–7 2–4 2–3

57 5–6

0.8

4–5 1–43–5 1–3 2–5 1–2

0.6

4–7 4–6 1–5 3–7 3–6

1–7 1–6

34

0.4

0.2

0

3

6

9

12

15 ω

Fig. 3. (a) Shape of the fewcycle pulse as a function of its absolute phase; (b) spectra of our incident (solid black line) and transmitted pulses ϕ0 = 0 (solid gray line), π/4 (dottedanddashed line), and π/2 (dashed line). Vertical dotted lines represent the transition frequencies.

the interaction between a pulse and a medium with seven energy levels. Some of these levels had rather close transition frequencies (e.g., 1–2, 1–3 and 1–4 or 3–5, 4–5 and 5–6); other parameters corresponded to earleir experiments: α = 67.54, dmn = 0.02, γmn = 0.5, Wmn = 10–7, ωp = 5, E0 = 1; and τp = 2. The results from calculations for three absolute phase values, ϕ0 = 0; π/4; and π/2, are shown in Fig. 3. Differences between their initial shapes can be seen in Fig. 3a, while their spectra are compared in Fig. 3b. It is obvious that the initial spectra of all three pulses are identical; however, the spectra of pulses after passing through the medium are different for different pulse

(9)

where ε(z) is the dielectric permeability of materials with different values in different layers. The number of layers in every structure was equal to 20, and the dielectric permeability of each layer could have values uniformly distributed over the range of 1 to 1.73. The thickness of the layers was equal to 0.25 dimensionless units, and the pulse wavelength was 0.2. The twolevel medium was characterized by transition frequency ω12 = 6, which corresponded to a wavelength of 0.1667 with dipole moment 0.3 and extension lmedium = 5. The results from our computer simulations are pre sented in Fig. 4. Figure 4a shows an example of a ran dom distribution of permittivity and the shape of the initial pulse; Fig. 4b shows the spectra of the incident and reflected pulses for the following cases: (1) a reso nant medium not packed in a disordered layered struc ture; (2) a resonant medium packed in the structure shown in Fig. 4a. It can be seen that the layered struc ture strongly distorts the reflected signal spectrum, leading in particular to the emergence of additional minima that can be interpreted as belonging to addi tional absorption frequencies that are missing in the substance. Figure 4c shows the spectra of incident and transmitted pulses for the same parameters. In con trast to the previous case, the spectrum of the trans mitted pulse is less distorted and the absorption line of the twolevel medium is still noticeable in the spec trum. However, the spectrum also contains other min ima that are due to the influence of the packing. In Fig. 4d, we can see the considerable influence the lay ered structure has on the reflected radiation spectrum (the reflected signal was the one used for identification purposes). Alongside the absorption line of the sub stance, the transmitted radiation spectrum contains other minima; this can lead to the false detection of materials.

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PROPAGATION OF FEWCYCLE PULSES (a)

2.0

Fig. 4. (a) Shape of the incident pulse (dashed line) and the schematic layout of our layered structures (solid line) and substance (horizontal dashed line); (b) spectra of the inci dent (dashed line) and reflected pulses for the medium placed between noncyclic layered structures (solid line), and for the medium without layered structures (dotted anddashed line). (c) Spectra of the incident (dashed line) and transmitted pulses for the medium placed between noncyclic layered structures (solid line), and for a medium without coating (dottedanddashed line). (d) Comparison of the spectra of the incident pulse (smooth solid line) and spectra averaged over a hundred measurements for the transmitted and reflected radiation of a medium with layered coating (dottedanddashed line and kinked gray solid line, respectively) and of layered structures with no substance between them (dashed and dotted lines).

1.5 1.0 0.5 0 –0.5 –1.0 –15 –10 –5

0

5

10

15

20 z

(b)

E(ω) 1.0

CONCLUSIONS The mechanism behind the cascade excitation of highenergy levels of molecules via intermediate energy levels when the energy of an emitted quantum is less than the difference between the energies of a particular transition was determined. It is because of this mechanism that the spectrum of a pulse transmit ted through the medium is enriched, relative to the spectrum of the incident pulse. The effect the absolute phase of an incident pulse has on its interaction with a medium was demon strated. It was shown in particular that the absolute phase affects the relative rates of transitions to nearby levels. It was shown that a disordered layered medium serving as packing for a medium can substantially dis tort the spectrum of a pulse transmitted through a sub stance, so we cannot detect a material by means of standard spectroscopy.

0.8 0.6 0.4 0.2 0 2

1511

4

E(ω) 1.0

6

8 ω

6

8 ω

(c)

0.8 0.6 0.4 0.2 0 2

4

ACKNOWLEDGEMENTS This work was supported by the Russian Science Foundation, project no.142100081.

(d)

E(ω) 1.0

REFERENCES

0.8 0.6 0.4 0.2 0 2

4

6

8 ω

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