Two-pulse photon echo area theorem in an optically dense medium

: We perform a theoretical and experimental study of the two-pulse photon echo area conservation law in an optically dense medium. The experimental properties of the echo signal are studied at 4K on the optical transition 3 H 6 (1) → 3 H 4 (1) (793 nm) of Tm 3 + in a YAG crystal for a wide range of pulse areas of the two incoming light pulses, up to θ 1 ≈ 4 π and θ 2 ≈ 7 π respectively, with optical depth 1.5. We analyze the experimental data by using the analytic solution of the photon echo area theorem for plane waves. We ﬁnd that the transverse Gaussian spatial proﬁle of the beam leads to an attenuation of the echo area nutation as function of θ 1 and θ 2 . Additional spatial ﬁltering of the photon echo beam allows to recover this nutation. The experimental data are in good agreement with the solution of photon echo pulse area theorem for weak incoming pulse areas θ 1 , 2 (cid:46) π . However at higher pulse areas, the observations diverge from the analytic solution requiring further theoretical and experimental studies. shows the photon echo pulse area as a function of the second incoming pulse area θ 2 ( 0 ) for diﬀerent optical depths of the medium α L . We examine the previously


Introduction
The optical analogue of the spin echo [1], the photon echo [2,3] is an efficient method in modern coherent spectroscopy [4]. It became a well-established tool to extract basic spectroscopic parameters of optical transitions and characterize fast atomic and molecular quantum dynamics up to the femtosecond timescale [5][6][7][8][9][10]. In applied research, the photon echo has been considered in the 70's as a tool for the fast storage and processing of the optical information [11][12][13][14][15][16]. More recently, it has been shown that modified schemes of photon echo in optically dense media can be efficiently applied for the quantum storage of photonic qubits [17][18][19][20][21][22][23], even if the use of strong π-pulses could be risky [24,25]. We take a widely used definition of optical density: the absorption coefficient multiplied by the sample thickness.
In this broad context, the McCall-Hahn pulse area theorem provides a global description of light pulse propagation in optically dense resonant atomic media [26]. The theorem can be seen as a general conservation law for the pulse area and a helpful complement to the energy conservation law. In essence, the area law is independent of the particular temporal shapes of the input pulses and is widely used as a descriptive parameter for spin and photon echo measurements. After 50 years, the theorem continues to attract researchers' attention [27][28][29][30][31][32][33][34]. Experimental studies of the pulse amplitude, spatial beam profile, temporal pulse shape and medium optical thickness on the photon echo intensity are usually described by the Maxwell-Bloch numerical simulation [35][36][37][38][39] or by using area theorem for a small input pulse area [40,41]. The most general case of the photon echo area theorem, i.e. valid for any arbitrary incoming pulse intensities, was also discussed in [42,43] in terms of total pulse area of all the generated signals. The analytical solution of the photon echo pulse area was obtained in [44] for arbitrary areas of the incoming pulses. Below we analyze these solutions and compare with new experimental results and existing numerical calculations [36].
It is worth noting that in the recent context of quantum information with the realization of optical quantum memories, the photon echo area theorem could provide general understanding of the signal propagation in optically dense samples [17,18,20,45]. The small input pulse area condition is not only a particular case of the area theorem but also a limitation for the quantum storage fidelity [24,25]. In any case, the recent progress in the extreme field confinement achieved for rare-earth ions embedded in photonic cavities should revive the interest in the area theorem since large areas can be obtained with only a few photons when the so-called strong confinement regime is targeted [46][47][48].
In the second section, we give a theoretical description of the photon echo pulse area theorem in an optically dense medium. In the next section, we present the experimental results obtained for a two-pulse photon echo sequence in a Tm 3+ :YAG crystal (c = 0.1 at. %). We vary the input pulse areas up to θ 1 (0) ≈ 4π and θ 2 (0) ≈ 7π for the first and second pulses respectively. The obtained experimental data are discussed in the light of the analytical solution for the photon echo area theorem, including the transverse Gaussian profile of the light beam. We partially retrieve the plane wave solution by using spatial filtering of the light beams.

Photon echo area theorem
We derive in this section the area theorem for the echo in a two-pulse photon echo sequence depicted in Fig. 1. We first remind the well-known McCall-Hahn area theorem for a single pulse. Following [26,27,44] and using the well-established Bloch-Maxwell formalism in one dimension (propagation along z) for two-level inhomogeneously broadened atomic ensemble, the pulse area θ 1 (z) follows the propagation equation known as the McCall-Hahn area theorem [26]: where α is the resonant absorption coefficient [42]. The solution of this equation is: An important consequence of Eq.
(2) is that if the incident pulse has a pulse area θ 1 (0) = kπ, k ∈ Z, then the area is conserved during the propagation. This is true for even values of k as noted in the original work [26] as well as for odd values of k. However, the latter case is much more difficult to implement experimentally, since the initial pulse area is preserved only when θ 1 (0) is very close or exactly equal to π, otherwise the pulse area converges towards the nearest even multiple of π while propagating.

Photon echo area theorem
Following the same approach, the theorem was extended to a two-pulse photon echo sequence. The pulse areas of second input pulse and photon echo follow the equations derived in [27] and [44]:  1. The classic two pulse echo sequence propagation through the Tm 3+ doped crystal. ì k is the common wave vector of three pulses parallel to the external magnetic field. The first signal pulse is partly absorbed by the medium, the second pulse triggers the rephasing process which results in the photon echo emission.
where θ i (z), i = 1, 2, e are the pulse areas of the first, second and echo pulses respectively, τ 12 is the delay between the incoming pulses, and T 2 is the coherence lifetime of the atomic transition. A detailed derivation of Eqs. (3) and (4) can be found in the Appendix.
The solution for the echo pulse area (6) is in agreement with the direct Maxwell-Bloch numerical simulations given in [36]. In particular the authors considered two cases: θ 2 (0) = π and θ 2 (0) = π/2 (T 2 τ 12 , θ 1 (0) = 0.27θ 2 (0), δt 1 > δt 2 ) and calculated the pulse energy efficiency as a function of optical depth αL. The calculated energy efficiency behavior coincides with the θ 2 e (z)/θ 2 1 (0) dependence depicted in Fig. 2. This agreement clearly shows the potential of the area theorem approach to quantitatively characterize the nonlinear interaction in optically dense media.
Let us now discuss the basic properties of the photon echo pulse area theorem in few particular cases.
These two cases satisfy the condition lim αz→∞ {θ 1 (z) + θ 2 (z)} = 2π(n + m), demonstrating an asymptotic formation of optical solitons after interacting with the second pulse. This situation actually corresponds to the McCall-Hahn theorem [26,42] due to the absence of photon echo in these cases (due to re-absorption).
Standard photon echo (π/2,π)-excitation This is the most popular case of the photon/spin echo spectroscopy with θ 1 (0) = π/2, θ 2 (0) = π (which gives β = 0, γ = ∞ in Eq. (6), and θ 2 (z) = π) so we can obtain the simplified solution: The echo pulse area is continuously amplified in the medium by the factor sinh( αz 2 ) with the asymptotic values for the gain: sinh αz 2 → 1 2 exp αz 2 and the echo area θ e (z) → π in the limit αz → +∞. In this limit, the total area of the three pulses θ 1 (z) + θ 2 (z) + θ e (z) → 2π, when the first pulse is completely absorbed and the area of the second pulse remains π. This also ensures the absence of any secondary echo pulse.

Weak first pulse (θ 1 (0)
π) This case typically corresponds to quantum memory schemes based on the photon echo in ensembles [19,50]. Assuming θ 1 (0) π, Eq. (6) leads to: An intense echo can also be retrieved when θ 2 (0) = π despite the small incoming pulse area. In that case, we obtain from Eq. (9): This expression shows that the echo pulse tends to a π-pulse in a sufficiently large optical depth. Area and energy efficiencies can be greater than unity [36,43,51]. This comes from pulse echo amplification in an inverted medium and leads to quantum noise when the photon echo is considered as a memory [25].
We see that initial condition θ 1 (0) 1 does not guarantee that the small echo pulse area θ e (z) 1 condition is maintained though propagation because of the echo amplification by the gain term sinh αz 2 . Equation (9) gives the appropriate condition for the weak first pulse approximation θ 1 (0) < 1/ sin 2 θ 2 (z) 2 sinh αz 2 at any position z ≤ L. In the special case θ 2 (0) = π and τ 12 T 2 , this translates into the condition θ 1 (0) < 1/ sinh αL 2 ≈ 2e −αL/2 leading to the solution of Eq. (9) [25]: Non-optimal photon echo generation (θ 2 (0) π) The echo pulse amplification can be suppressed by the spatially dependent factor sin 2 (θ 2 (z)/2) for z > 2β/α if the input pulse area of the second light pulse θ 2 (0) π. For sufficiently large input pulse area of the first pulse θ 1 (0) > 1 in Eq. (6) and θ 2 (0) π, we have the following asymptotic decrease of the echo pulse area: Equation (12) shows that echo signal is generated and propagates only at the entrance of the atomic medium. We also find in Eq. (12) that an optical soliton is formed at z/α 1 [26,42] if θ 2 (0) > π. A wide range of interesting physical effects associated with the soliton formation is however beyond the scope of this paper. Equation (12) shows that in optically dense media, the echo pulse area can take any values in the interval [−π, π] starting from zero, it peaks at some value of z and then decays to zero with z → ∞. The fact that area of each echo pulse is less than π does not contradict with the McCall-Hahn area theorem [26,40] predicting 2π pulse formation, when θ 1 (0) + θ 2 (0) > π. It only means that one needs to consider also higher order echos (after the primary echo) which are also exciting the medium.
In summary, Fig. 3 shows the photon echo pulse area as a function of the second incoming pulse area θ 2 (0) for different optical depths of the medium αL. We examine the previously discussed cases: Fig. 3(a) and θ 1 (0) = π/2 in Fig. 3(b). For a given optical density αL, the photon echo pulse area θ e (θ 2 (0)) undergoes a series of oscillations peaking at θ 2 (0) = (2n + 1)π. It is interesting to note that the positions of the maxima are independent of θ 1 (0). The peak widths, however, become sharper at higher optical densities. The insets in Figs. 3(a) and 3(b) show the cross sections at αL = 2 and αL = 10 and reveal these tendencies. The non-linear nature of the process in an optically dense (αL 1) medium causes the echo to form practically only when the second incoming pulse area is an odd multiple of π. For θ 2 (0) (2n + 1)π, the echo area is maximum at a specific optical depth and then decreases with the increase of αL. On the contrary, for θ 2 (0) = π the echo area asymptotically tends towards π which could lead to area efficiency |θ e (L)/θ 1 (0)| 2 > 1 only limited by finite optical density αL, relaxation time T 2 and non-ideality of π-pulse.
Below we experimentally test the photon echo area theorem in more general conditions, namely high incoming pulse areas θ 1 (0) > π, θ 2 (0) > 2π and different spatial filtering profiles.

Including the transverse Gaussian beam profile
In our experiments, the light beams have a Gaussian profile so the field in the atomic medium is described by E(t, r) = E 0 (t)e −(r/r 0 ) 2 where r is the transverse coordinate and r 0 the beam radius. A complete study of the Maxwell-Bloch equations in two dimensions z and r is a complex problem well beyond the scope of this work. Indeed, the two-level dynamics induces additional highly non-linear terms in the field master equation (paraxial Helmholtz equation with a source term). We extend in a simple manner the photon echo area Eqs. (1), (3) and (6) derived for plane waves to include the transverse spatial beam profile and give a quantitative explanation of the experimental data. For the sake of simplicity, we assume the geometrical optics approximation i.e. neglect diffraction effects (similar to the studies of the resonators stability presented by Siegman in [52]). This assumption is valid for a sufficiently large beam cross-section πr 2 0 Lλ (λ is the wavelength of light). However, it should be noted that in our experimental conditions, the inequality is not fulfilled (πr 2 0 = 0.3Lλ) to favour large pulse areas. So diffraction effects could play a considerable role in our case.
Experimentally, the medium is imaged onto a detector with a magnification factor M. The measured detector signal is proportional to the field intensity averaged over the active detection surface. This latter can be connected to the theoretically calculated θ e (r, L) (see Eq. (13)) by where r m is the cutoff radius and S = πr 2 m the surface of collection. The measurement depends on the spatial filtering procedure (pinhole in our case) and the local echo pulse area θ e (r, L).

Experimental setup
The crystal under investigation is 0.1 at.%Tm 3+ :Y 3 Al 5 O 12 (YAG) that has been well studied as an optical quantum memory medium [53][54][55] due to the 3 H 6 (1) → 3 H 4 (1) transition at 793 nm. The inhomogeneous broadening is ∆ inh ≈ 20 GHz. The 2 × 3 × 8 mm 3 crystal is cooled down to 4K in a Montana Instruments cryostat. The optical density αL along the 8 mm long dimension reaches 1.5 at the central frequency. The crystal is placed in a moderate 600 G magnetic field pointing along the [001] crystal axis. The field is provided by two pairs of 25 mm permanent NdFeB magnets (at 30 mm from the crystal). In this configuration, the optical transition coherence lifetime is increased to T 2 = 189 µs.
The light beams are generated by a single frequency continuous Ti:Sapphire laser. The beams pass through an acousto-optic modulator (AOM1) that temporally shapes the incoming pulses of the two-pulse photon echo sequence. We use a single mode fiber for mode filtering. Lens L1 focuses the light on the crystal with a waist r 0 = 24.3µm. Light propagates along the [110] crystallographic axis and is polarized along the [001] crystalline axis. The details of the mutual orientation of the crystal axes and electric and magnetic fields are shown in Fig. 1. The crystal is imaged outside the cryostat with a 75 mm lens L2, with a magnification factor M = 4. This results in a Gaussian beam with waist r p = Mr 0 = 97 µm in the crystal image plane.
To study the effect of the spatial beam profile on the photon echo area theorem we perform two types of experiments, without spatial filtering (NoSF-experiment) and with spatial filtering (SF-experiment). With these two experiments, we evaluate how different degrees of spatial inhomogeneity affect the characteristic nutation of the detected echo pulse depending on the incoming pulse areas θ 1 (0) and θ 2 (0). In the NoSF-experiment, the detector is directly placed in the crystal image plane (see Fig. 4). In the SF-experiment, analogous to the works [30,36], a 50 µm pinhole (r m = 25 µm) is placed in the crystal image plane. This pinhole selects the most intense light field coming from the center of the Gaussian profile such that E(t, r m ) = 0.93E(t, 0). Then the lens pair L3 and L4 focuses the light transmitted by the pinhole onto AOM2 that temporally gates the echo signal and thus isolates the avalanche photodetector from the intense excitation pulses. A higher sensitivity avalanche photodetector is used for the SF experiment to compensate for the lower signal obtained after spatial filtering. Lenses L5 and L6 focus the light on the detector.

McCall-Hahn area theorem verification
We start by verifying the McCall-Hahn area theorem for a single pulse excitation. For this purpose, we carry out the experiment with spatial filtering (r p = 97µm and r m = 25µm) to be as close as possible to the plane wave case. The pulse duration is δt = 2µs and we vary the pulse amplitude to control the input area θ 1 (0, 0).

The obtained experimental dependence of the normalized value
is the detector signal) on the pulse area of input signal is depicted as black squares in Fig. 5. The experimental data A D are in good agreement with theoretical curve θ 2 e (r m ) 1/2 from Eq. (14) obtained in [26].
To fit the experimental data we used two fitting parameters, the scaling coefficients for x and y axis.

Photon echo area theorem with Gaussian beams
In the NoSF-experiment, the entire beam going through the sample is collected on the detector. In this experiment no external magnetic field is applied to the crystal so the coherence lifetime T 2 = 76µs is shorter, optical density at the line center is αL = 1.5, with α = 187.5 m −1 and L = 8 mm. We vary the amplitude of the second light pulse (and thus the input area θ 2 (r, 0)), while maintaining constant the first pulse area θ 1 (0, 0). The pulse durations are δt 1 = δt 2 = 2µs and the delay between the pulses is τ 12 = 8µs. Fig. 6 shows the experimental data for three different input areas of the first pulse: θ 1 (0, 0) ≈ 0.3π, 0.7π and 1.4π. The nutation behavior of the echo signal as a function of θ 2 (0, 0) is significantly damped for the first and second cases (see black squares and red circles in Fig. 6) and almost absent in the third case of the highest θ 1 (0, 0) ≈ 1.4π (see blue triangles in Fig. 6). This is due to averaging over the transverse spatial profile of the detected echo signal as U D (t) ∼ ∫ ∞ 0 I e (r, t)r dr, since r m r 0 (see Eq. (14)). Figure 6 also shows the theoretical calculations of the photon echo signal characterized by the value θ 2 e (r m ) 1/2 (see Eq. (14)) in these experimental conditions. To obtain these curves, we use only two fitting parameters, namely two scaling factors for the horizontal (absolute value of the incoming pulse area) and vertical axis (normalized value A D ). The horizontal scaling factor is proportional to the experimentally measured Rabi frequency which in this experiment was ≈ 320 kHz, which is approximately 3 times lower than in the SF-experiment.
The experimental and theoretical dependencies in Fig. 6 are in a good agreement for the small incoming pulse area θ 2 (0, 0) < 1.3π for all three curves with different values of pulse area Output signal, a.u.
is the detector signal) and theoretical (red line, θ 2 e (r m ) 1/2 from Eq. (14)) dependences of a single pulse area after the resonant medium as a function of the incoming pulse area, corresponding to the McCall-Hahn area theorem. θ 1 (0, 0) = 0.3π, 0.7π, 1.4π. As expected, the experimental behavior of echo signal deviates from the theoretical solution Eq. (14) at higher pulse area θ 2 (0, 0). One reason for this is that the experimentally detected pulse intensities cannot be expressed in terms of pulse areas for θ 1 (0), θ 2 (0) > π due to the transverse oscillating spatial behavior. Moreover, we observe an increased optical nutation period at higher input pulse area θ 2 (0, 0) as compared to the theoretical curves. This non-obvious nonlinear effect of multi-pulse excitation deserves a special study and is beyond the scope of this paper. Perspectives are discussed in the conclusion.

Photon echo area theorem with spatially filtered beams
To retrieve the echo nutation, we perform the SF-experiment using a pinhole for the detection as depicted in Fig. 4 with estimated Gaussian beam radius r p = 97µm at the pinhole location. We realize two sets of measurements by collecting the echo signal for varying incoming pulse areas θ 1 (0, 0) and θ 2 (0, 0). In this section, the experimental parameters are r p = 97µm, r m = 25µm, L = 8mm, α = 187.5m −1 , T 2 = 189µs, the measured Rabi frequency of the transition Ω ≈ 900 kHz. They are also used for the numerical calculations from Eqs. (6) and (14). The pulse durations are δt 1 = 5µs, δt 2 = 2 µs and the time delay between the exciting pulses is τ 12 = 20µs. Figure 7 shows the experimental and theoretical echo signal dependencies as a function of the second pulse area for two different values of the first pulse input area. Again, we also use two fitting parameters (scaling factors for both axis). The horizontal scaling factor is the same as for single pulse dependence in Fig. 5. Each pair (experimental data set and theoretical curve) corresponds to a given value of the first pulse incoming area θ 1 (0, 0). Figure 7 shows that spatial filtering reveals the nutation and significantly reduces the damping. The particular case of θ 1 (0, 0) < 1 was experimentally studied earlier in [30] for θ 2 (0, 0) < 2π where the obtained results were satisfactorily described by the plane wave solution (Eq. (9)).
We note that both theoretical curves and experimental data, exhibit a series of damped Normalized echo area, a.u.

Fig. 6. Three experimental (symbols, normalized value
is the detector signal) and the corresponding theoretical (lines, normalized value ∼ θ 2 e (r m ) 1/2 from Eq. (14)) dependencies of the photon echo pulse signal as a function of the incoming area of the second pulse. Each set corresponds to a different initial area of the first incoming pulse. Black squares and black line correspond to θ 1 (r = 0, z = 0) = θ 1 (0, 0) ≈ 0.3π, red circles and red line to θ 1 (0, 0) ≈ 0.7π, blue triangles and blue line to θ 1 (0, 0) ≈ 1.4π.
oscillations. The agreement is quite accurate at low input pulse area θ 2 (0, 0) < 2π. However, at higher θ 2 (0, 0), the experimental curves are strongly damped as compared to the theoretical prediction. The curves are stretched in the large area region showing a clear non-harmonic behavior. The increase of the nutation period is an interesting feature that we see much more clearly than in the NoSF-experiment. This general discrepancy indicates the limitations of the geometrical optics approximation that we assumed in averaging over the beam profile. This issue will be also discussed in the conclusion.
We also study the experimental echo signal (black squares in Fig. 8) as a function of the first pulse area θ 1 (0, 0). The solid line is the theoretical fit (see Fig. 8) following a similar procedure to Fig. 7. As before, we plot the theoretical value θ 2 e (r m ) 1/2 (from Eq. (14)) and the experimental data as ∫ dt √ U D for constant θ 2 (0, 0) = 1.1π. Again, the theory coincides with the observed data at small input pulse area θ 1 (0, 0) < π and only qualitatively describes the nutation at higher θ 1 (0, 0) values where the observed oscillations are strongly damped.
Both dependencies of echo signal on θ 2 (0, 0) and θ 1 (0, 0) demonstrate quite similar features for increasing θ 2 (0, 0) and θ 1 (0, 0). Namely, we observe the oscillation stretching of the experimental curve (longer periods at higher areas). The similar pattern indicates common physical effects in the observed nutation taking place in an optically dense medium.

Discussion and conclusion
We have performed theoretical and experimental studies of two-pulse photon echo in the optically dense medium. We have used the optical transition 3 H 6 (1)→ 3 H 4 (1) (793 nm) of a Tm ion-doped YAG crystal cooled at 4K. Herein, the echo area has been studied as a function of incoming pulse areas of the first θ 1 (0, 0) and second θ 2 (0, 0) pulses up to 4π and 7π, respectively. We have observed that inhomogeneous beam intensity profile leads to the strong damping of the expected  To analyze the experimental data we have developed a theoretical approach based on the photon echo area theorem. We applied the theorem for two-dimensional Maxwell-Bloch model with spatial transverse inhomogeneity of the light beams considered in the geometrical optics approximation. This model satisfyingly describes the data for relatively small incoming pulse areas θ 1 (0, 0), θ 2 (0, 0) < 2π and provides a qualitative agreement of the observed echo signal nutation at higher input pulse areas. In this region the theorem also predicts the energy efficiency of the echo in agreement with Maxwell-Bloch simulations presented in [36].
We also note that the geometrical optics approximation cannot describe the self-focusing and de-focusing of the beams propagating in an optically dense and spectrally selective atomic medium. Including the non-linear diffraction in the paraxial Helmholtz equation is a complex analytical task which is beyond the scope of this work. Atomic dipole-dipole interactions, as instantaneous spectral diffusion [56][57][58], cannot be also completely discarded.
Finally, the photon echo pulse area theorem provides a useful and general tool to study the light-atom interactions [59][60][61][62] revealing fundamental processes that play a role in many quantum optical memory protocols realized in the optically dense media [19,[63][64][65][66].
The formal solution of Eqs. (16) and (17) can be written as: dt Ω(t , z)w(t , z, ∆)e −(i∆+γ)(t−t ) . (19) Substituting the solution for P 12 (t, z, ∆) in Eq. (15) and integrating it over the time period of the specific pulse [t 0 , t 0 + τ 12 ], (where t 0 = t e − τ 12 /2, t e is the time of the echo emission, τ 12 is the time delay between the pulses) we get: dt Ω(t , z) w(t , z, ∆)e −(i∆+γ)(t−t ) (20) We are especially interested in the formation of a primary photon echo with emission time t e = 2τ 12 in the limit of a sufficiently large time delay between the two incoming light pulses τ 12 δt i (where δt i is a temporal duration of the i-th input light pulses). After the action of the two light pulses, the atomic variables P 12 and w acquire several components characterized by different spectral properties. Two components of the coherence P 12 describe the dephasing of the atomic coherence P 1 e −i(∆−iγ)t and P 2 e −i(∆−iγ)(t−τ 12 ) (the free induction decay terms after the first and the second pulses) and the only component characterizing the coherence rephasing determining the emission of the primary echo P 3 e −i∆(t−2τ 12 ) e −γt at time t = 2τ 12 . In the limit of ∆ → 0, these atomic coherences take the forms: P 1 → −i sin θ 1 (z) cos 2 θ 2 (z)/2, P 2 → −i cos θ 1 (z) sin θ 2 (z) and P 3 → (−i)v e (t 0 , z, 0) = −i sin θ 1 (z) sin 2 θ 2 (z)/2 [3,42]. The echo field is irradiated initially by the rephasing polarization P 3 and highly modified during propagation by the acquired atomic inversion w.
Here we assume that the input light pulses are well separated with the time delay between pulses being much larger than the temporal duration of the light pulses. With the averaging over inhomogeneous broadening of atomic transition ... in Eq. (15), only those components of polarization P 12 and w will participate in the echo formation due to their rephasing at the time of the echo emission. Namely the terms P 3 and w 1 affect the echo field emission while the other terms disappear after the averaging over the inhomogeneous broadening in the limit τ 12 δt e (where δt e is a temporal duration of the echo pulse δt e ∼ δt). Taking into account these properties of different spectral components in the averaged polarization and inversion, we can write for the averaged atomic coherence P 12 (t 0 , z, ∆)e −(i∆+γ)(t−t 0 ) | echo e −2γτ 12 P 3 (t 0 , z, ∆)e −i∆(t−2τ 12 ) , w(t 0 , z, ∆) w 1 (t 0 , z, ∆) .