Ultracold plasma (UCP) was first obtained in Ref. [1] where the method of near-threshold two-stage laser ionization proposed in Ref. [2] was used to turn laser-cooled ⁸⁸Sr atoms into the state of UCP. In such a plasma, the ion temperature T_i varies from 10⁻² K to 1 K, the electron temperature T_e ranges from 1 K to 1000 K and the concentration of charged particles is n ≤ 10¹⁰ cm⁻³.^[3] The availability of such a medium opens up unique prospect for studying macroscopic and microscopic processes in ultracold plasma. In Refs. [4] and [5] a method of cooling ions was proposed aiming at obtaining non-ideal ion-ion interaction in electron-ion plasma. Later it was shown that the nonlinear dependence of laser friction force on the velocity of ions is an important consideration for laser cooling^[6] and the spectrum of cooling laser radiation was described.^[7]

Photoionization of atoms induces ion–ion interaction resulting in correlation heating of ions.^[8] Absorption spectroscopy is one of the instruments to study gases.^[9] In this paper we use this instrument to study spectral characteristics of absorption (optical thickness and line shape and width) in UCP under correlation heating and compare our results with the experimental findings.^[3,10,11]

Formation of UCP starts with two-stage photoionization of an ensemble of cooled atoms having spherical geometry; therefore the initial space distribution of ions is a replica of atomic distribution and is described by the following expression:^[3]

Radiation absorption I(v,r) obeys the Beer–Bouger–Lambert law

The shape of absorption profile in such a plasma is determined by Lorentz broadening resulting from spontaneous decay and Doppler broadening due to thermal motion of ions. Under the joint influence of these broadenings the absorption coefficient has the Voigt profile^[12] and, considering Eq. (1), will be given by

Considering that k₀ and a depend on V₀, we rewrite expression (7) as

The Voigt profile turns into the Lorentz profile for large c, and the absorption coefficient becomes

To analyze the situation, we take the origin of the coordinate system to be the center of the sphere. Then for the Voigt profile, using expression (10), we write an expression for the optical thickness along the diameter when ν = ν₀, with assuming T_i to be constant over space, as follows:

Our calculations are based on the experimental data for strontium ions:^[3,10,11] λ₀ = 421.7 nm (²S_1/2–²P_1/2 transition), γ/2π = 21 MHz, γ_L/2π = 5 MHz, n₀ = 2· 10¹⁰ cm⁻³, σ = 0.6 mm, initial T_i = 10⁻² K, T_e = 56 K. Note that for the given parameters .

The variations of optical thickness with a derived from expressions (13)–(15) for various types of broadenings, are shown in Fig. 1. The results obtained indicate that the optical thickness for Voigt broadening is the same as that for Doppler broadening if a < 0.06 and as that for Lorentz broadening if a = 3. The optical thickness for 0.06 < a < 3 is affected by both Lorentz and Doppler broadening.

Fig. 1.

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	Fig. 1. Plots of optical thickness versus a for various types of broadenings: 1: Doppler broadening (15), 2: Lorentz broadening (14), 3: Voigt broadening (13).

According to the numerical data (Fig. 1), the optical thickness τ₀ varies from 0.65 for a = 3.9, T_i = 10⁻² K to 0.26 for a = 0.39, T_i = 1 K. Due to correlation heating the ion temperature grows linearly with time increasing from T_i = 10⁻² K until it reaches its maximum T_i ≃ 1 K at t ≃ 0.3 μs.^[8] Hence T_i ≃ 0.3 K, a ≃ 0.7 at t ≃ 0.1 μs. Then for a ≃ 0.7 from Fig. 1 we have τ₀ = 0.36. The experimental optical thickness under these conditions is τ₀ = 0.28.^[3,10] However, considering the ±40% accuracy of the measured ion concentration n₀,^[3] the error for the optical thickness will be Δτ₀ ≈ ± 0.1, which means that the experimental and calculated data are in good agreement.

To find an integral optical thickness τⁱ, we will use a sphere of radius . Consider the propagation of radiation along the chord parallel to the vector of the direction of propagation e. The chord location is determined by the angle φ from the sphere center between the direction e and the endpoint of the chord on the surface of the sphere (for φ = 0, radiation propagates along the sphere diameter). So

Figure 2 shows the integrated optical thickness as a function of a. The experimental value for t = 75 nsec equals 0.8 mm².^[11] For this timescale, T_i ≃ 0.25 K, a ≃ 0.8; and hence from expression (18) and Fig. 2 we have mm², which agrees with the experimental value.

Fig. 2.

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	Fig. 2. Optical thickness integrated over sphere for the Voigt profile.

The absorption line profiles for various types of broadenings are given by expressions (10)–(12) and can be seen in Figs. 3–5. Here Δν = ν − ν₀. Plotted on the Y-coordinate are absorption coefficients for Δν = 0 divided by .

Fig. 3.

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	Fig. 3. Absorption line profiles (a = 0.01), solid line Voigt broadening, ◊: Doppler broadening.

Fig. 4.

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	Fig. 4. Absorption profiles (a = 3.9), ◊: Lorentz, solid line: Voigt.

The Voigt and Doppler profiles coincide with each other for small a (Fig. 3) while the Lorentz and Voigt absorption curves coincide with each other for large a (Fig. 4). The absorption line profile for any a in between these values is given by the Voigt formula (Fig. 5).

Fig. 5.

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	Fig. 5. Absorption profiles (a = 1), ◊: Lorentz, o: Doppler, solid line: Voigt.

The temperature T_i and a being independent of the coordinates, the absorption line profile for radiation propagating along the sphere diameter will coincide with the one integrated over sphere. It should be noted that the frequency dependence of the absorption coefficient is an important factor to be considered when dealing with laser cooling of ions.^[7]

The spectral width of the absorption line Δν equals double the value Δν = ν₁ − ν₀, where ν₁ is the frequency at which the absorption is half the absorption in the center of the line at ν = ν₀.^[12] So the absorption line spectral width is

The variations of line width with a for various types of broadening are shown in Fig. 6. It is clearly seen that the Lorentz and Voigt line widths coincide with each other for a = 3 and the Doppler and Voigt line widths coincide with each other when a < 0.06. For t = 75 ns, the experimental value is Δν ≈ 62 MHz^[3,11] and the calculated one is 52 MHz.

Fig. 6.

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	Fig. 6. Absorption line widths versus a for different broadenings, 1: Doppler broadening, 2: Lorentz broadening, 3: Voigt broadening.

In this paper, we deal with the absorption of laser radiation in UCP under correlation heating of ions. The spectral characteristics (optical thickness and the line shape and width) are calculated for the absorption coefficient given by the Voigt formula. The spectral data for Doppler and Voigt broadenings are shown to coincide with each other if a < 0.06, and the spectral data for Lorentz and Voigt broadenings also coincide with each other when a = 3. The increase in the ion temperature due to correlation heating results in a reduced optical thickness and increased absorption line width. The numerical estimates obtained are in good agreement with the experimental results reported in Refs. [3], [10], and [11].