ISSN 0021-3640, JETP Letters, 2015, Vol. 102, No. 8, pp. 487–492. © Pleiades Publishing, Inc., 2015. Original Russian Text © A. Sargsyan, Y. Pashayan-Leroy, C. Leroy, Yu. Malakyan, D. Sarkisyan, 2015, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2015, Vol. 102, No. 8, pp. 549–554.
OPTICS AND LASER PHYSICS
Features of Faraday Rotation in Cs Atomic Vapor in a Cell Thinner than the Wavelength of Light1 A. Sargsyana, *, Y. Pashayan-Leroyb, **, C. Leroyb, ***, Yu. Malakyana, and D. Sarkisyana a Institute b Laboratoire
for Physical Research, National Academy of Sciences of Armenia, Ashtarak-2, 0203 Armenia Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303-Université de Bourgogne-Dijon, France * e-mail:
[email protected] ** e-mail:
[email protected] *** e-mail:
[email protected] Received July 27, 2015
Features of the effect of Faraday rotation (the rotation of the radiation polarization plane) in a magnetic field of the D1 line in Cs atomic vapor in a nanocell with the thickness L varying in the range of 80–900 nm have been analyzed. The key parameter is the ratio L/λ, where λ = 895 nm is the wavelength of laser radiation resonant with the D1 line. The comparison of the parameters for two selected thicknesses L = λ and λ/2 has revealed an unusual behavior of the Faraday rotation signal: the spectrum of the Faraday rotation signal at L = λ/2 = 448 nm is several times narrower than the spectrum of the signal at L = λ, whereas its amplitude is larger by a factor of about 3. These differences become more dramatic with an increase in the power of the laser: the amplitude of the Faraday rotation signal at L = λ/2 increases, whereas the amplitude of the signal at L = λ almost vanishes. Such dependences on L are absent in centimeter-length cells. They are inherent only in nanocells. In spite of a small thickness, L = 448 nm, the Faraday rotation signal is certainly detected at magnetic fields ≥0.4 G, which ensures its application. At thicknesses L < 150 nm, the Faraday rotation signal exhibits “redshift,” which is manifestation of the van der Waals effect. The developed theoretical model describes the experiment well. DOI: 10.1134/S002136401520014X
Magneto-optical effects, in particular, linear and nonlinear Faraday rotation effects, are excellent tools for laser spectroscopy of gases [1–4]. Optical magnetometers based on an ultranarrow spectral width with large polarization rotation are actively used. Spectroscopic cells with lengths of 1–10 cm filled with alkali metal vapors are basic elements of these types of optical magnetometers. The creation of narrowband optical atomic filters based on Faraday rotation is promising because they involve crossed polarizers and radiation of only the desired signal is transmitted [5]. Furthermore, the transmission spectrum of such filters, in particular, on the D1 line of Cs atoms, can be three orders of magnitude narrower than that for interference filters [6]. The authors of [7] studied Faraday rotation for 87Rb atoms in high magnetic fields at which the coupling between the total angular momentum of an electron J and the magnetic moment of the nucleus I begins to disappear and the splitting of atomic levels is described by the projections mJ and mI (the so-called Paschen–Back effect on the hyperfine structure [8]). The authors of [9] obtained the image 1 See
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of domains of magnetic strips of credit cards using thin iron-doped garnet films and Faraday rotation. It was previously shown that cells with a nanometer thickness in the direction of propagation of laser radiation (with two other centimeter dimensions) have two highlighted thicknesses of an atomic vapor column: L = λ/2 and λ [10–12]. In the former case, the absorption and fluorescence spectra demonstrate a significant sub-Doppler narrowing (by a factor of 3 and 6, respectively) compared to centimeter-length cells. In the latter case (L = λ), the absorption spectrum is only a factor of 1.3 narrower as compared to centimeter cells (in this case, the fluorescence spectrum is narrower by a factor of 3). However, the main feature is the formation of so-called atomic-velocity selective optical pumping resonances (VSOPs) in the absorption spectrum, which correspond to atomic transitions with the linewidth close to natural [13]. This work is the first study of the features of the Faraday rotation effect for the D1 line in Cs atomic vapor in a nanocell with the thickness L varying in the range of 80–900 nm. It is shown experimentally and theoretically that the minimum spectral width and the maximum angle of rotation of the polarization plane in a longitudinal magnetic field are reached at the
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Fig. 1. (Color online) Layout of the experimental setup: (ECDL) diode laser, (FI) Faraday insulator, (G1, 2) Glan polarizers, (1) diode laser, (2) nanocell with Cs inside the oven, (3) Helmholtz coils, and (4) photodetector; φ is the angle of rotation of the polarization plane (in the main text, it is denoted as φF). The upper left panel shows the scheme of the levels involved in the Cs D1 line. The upper right panel is the photograph of the nanocell filled with Cs, where interference fringes formed at the reflection of light from the inner surfaces of the windows are seen; ovals mark the regions L = λ/2 = 448 nm and L = λ = 895 nm.
thickness of the column L = λ/2 = 448 nm. It is also shown that the spectrum of the Faraday rotation signal at the thickness of the Cs vapor column L < 150 nm acquires “redshift,” which is a manifestation of the van der Waals effect. It is important to note that, as was previously shown, atoms moving parallel to the windows of a nanocell, i.e., perpendicular to the laser beam, make the main contribution to the formation of signals of fluorescence, absorption, electromagnetically induced transparency, etc. The reason is that the time of their interaction with the laser beam is about four orders of magnitude larger and the number of inelastic collisions with the windows is several orders of magnitude smaller than those for atoms flying along the beam [10–12]. These atoms make the main contribution to the Faraday rotation signal. Figure 1 shows the layout of the experimental setup for recording the spectrum of the Faraday rotation signal with (2) a nanocell filled with Cs and radiation from (1) a cw narrowband extended cavity diode laser (ECDL) with a wavelength of 895 nm and a linewidth of ~1 MHz. The nanocell was placed at the center of pairs of Helmholtz coils 3, which compensated the Earth’s magnetic field and created a magnetic field in
the necessary direction, in particular, the longitudinal magnetic field B in the direction k of propagation of laser radiation. In order to form a frequency reference, a fraction of laser radiation was guided to an additional nanocell with the length L = λ, where VSOPs were formed at the 4 → 3', 4' transitions (prime marks the upper levels), which were used as the frequency reference [13]. The upper left panel in Fig. 1 shows the scheme of the levels involved in the D1 line of Cs atoms. The Faraday rotation signal was studied at the 4 → 3' transition. An advantage of the chosen scheme of the levels (Cs D1 line) is the largest frequency shift (1.168 GHz) among all alkali metals between the elements of the hyperfine structure of the upper levels, which makes it possible to study an individual atomic transition. A photograph of the nanocell filled with Cs is shown in the upper right panel in Fig. 1. The windows of the nanocell were fabricated from well-polished crystalline sapphire and had dimensions of 20 × 30 mm and a thickness of 2 mm. To minimize birefringence, the C axis was perpendicular to the surface of the window. The regions with the thicknesses L = λ/2 = 448 nm and L = λ = 895 nm are marked by ovals in the figure. A thin sapphire side-arm (SA) filled with JETP LETTERS
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Fig. 3. (Color online) Theoretical spectra of the Faraday rotation signal at the thicknesses of the nanocell L = λ/2 and λ; B = 5 G. The Rabi frequency of the laser field Ω = 0.1 MHz. The inset shows the theoretical transmission spectra of the nanocell at L = λ/2 and λ.
Fig. 2. (Color online) Spectra of Faraday rotation signals at the output of the second crossed polarizer G2 at the thicknesses of the nanocell L = λ/2 = 448 nm and L = λ = 895 nm; PL = 5 μW, B = 5 G, and TSA = 140°C. The inset shows the transmission spectra of the nanocell at L = λ/2 and λ.
metallic Cs is seen in the lower part. The SA of the nanocell was heated in the experiment in the temperature range of 120–230°C. The nanocell allowed heating up to 450°C (additional details were presented in [14, 15]). Figure 2 shows the spectra of Faraday rotation signals at the output of the second crossed Glan polarizer G2 (it is also called an analyzer) at thicknesses of the nanocell L = λ/2 = 448 nm (FWHM 35 MHz) and L = λ = 895 nm (140 MHz). It is noteworthy that a change in the thickness L by 5% weakly affects the spectra. The inset shows the transmission spectra at thicknesses of the nanocell L = λ/2 and λ (radiation power PL = 5 μW, magnetic field 5 G, and temperature of the appendix TSA = 140°C). As is seen in the figure, although the thickness is halved, the amplitude of the Faraday rotation signal at L = λ/2 is a factor of 3.5 larger than that at L = λ. This is in good agreement with the theoretical curve (Fig. 3) plotted within the model [16] for the Rabi frequency of the laser field Ω = 0.1 MHz (the expression for the recalculation of PL to Ω can be found in [16]). The results obtained can be qualitatively explained by the following simple expressions. The radiation power at the output of the polarizer G2 (crossed with the polarizer G1) can be represented in the form
Pout = Pin (1 − α) 2 sin 2 ϕ F ,
(1)
where φF is the angle of rotation of the polarization plane and α is the absorption coefficient. In our case, α < 0.01 [9] and Eq. (1) is reduced to Pout ≈ Pin ϕ F2 . JETP LETTERS
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In the case of linear Faraday rotation, the angle of rotation is (2) ϕ F ≈ 2 g Fμ BBL/ Γ L0 , where gF is the Landé factor, μB is the Bohr magneton in units of ℏ, B is the magnetic field, L is the thickness of the vapor column, Γ is the linewidth of the atomic transition, and L0 is the reduced thickness [1]. Expression (2) is valid under the condition gFμB ≪ Γ, which is satisfied in our case to magnetic field B < 50 G. Then, (3) ϕ F2 ~ (BL/ Γ) 2 . In the inset of Fig. 2, where the transmission spectra at thicknesses L = λ/2 = 448 nm and L = λ = 895 nm are shown, the width of the spectrum is Γλ/2 ≈ 110 and 300 MHz at L = λ/2 and λ, respectively; i.e., the spectrum at L = λ/2 is about three times narrower. Consequently, the reduction of L by half is compensated by the narrowing of the transmission spectrum at L = λ/2 [10]. According to Eq. (3), the Faraday rotation signal at L = λ/2 should be at least twice as large. This difference in the behavior of the Faraday rotation signal becomes more significant at an increase in the power of the laser >0.5 mW. Figure 4 shows the spectra of the Faraday rotation signal at the thicknesses of the nanocell L = λ/2 and λ and at a radiation power of 0.84 mW. In this case, for the thickness of the vapor column L = λ, strong optical pumping [17] transfers atoms from the Fg = 4 level to the ground Fg = 3 level, thus reducing the population of the Fg = 4 level. This reduction strongly suppresses the Faraday rotation signal at exact resonance with the 4–3' transition. Optical pumping forms VSOPs in the transmission spectrum of the nanocell with L = λ, whereas VSOPs do not appear for the nanocell with L = λ/2 at intensi-
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Fig. 4. (Color online) Spectra of Faraday rotation signals at the same parameters as in Fig. 2, but at PL = 0.84 mW. The optical pumping effect observed at L = λ strongly suppresses the Faraday rotation signal at exact resonance. The inset shows the dependence of the signal on PL at L = λ/2 (the dashed line is given for convenience).
Figure 6 shows the magnetic field dependence of the Faraday rotation signal. The magnetic field dependence of the signal (i.e., φ2) up to magnetic fields of ~7 G is almost quadratic in agreement with Eq. (3). At B = 5 G, φ ≈ 10 mrad, which is more than five times larger than a similar value from [9]. At B = 0.4 G, the Faraday rotation signal was still detected without special techniques, whereas the minimum magnetic field necessary to detect Faraday rotation in [9] was 5 G. It is worth noting that a small spectral width of the Faraday rotation signal will make it possible to improve the sensitivity of the Faraday rotation signal to the magnetic field by several orders of magnitude when using the synchronous detection technique. This will allow the creation of a magnetometer with a submicron spatial resolution for measurements of strongly inhomogeneous (gradient) magnetic fields. Figure 7 shows the dependence of the spectrum of the Faraday rotation signal on the temperature of the side-arm in the range of 119–227°C, which corresponds to the variation of the density of atoms in the range of 4 × (1013–1015) cm–3. It is seen that, at a small increase in the amplitude of the signal, its spectrum is strongly broadened (from 35 to 200 MHz). The inset of Fig. 7 shows the temperature dependence of the spectral width of the signal. The optimal temperature at which the amplitude is maximal at the minimum spectral width is ~150°C. The observed spectral self-broadening is due to collisions between Cs atoms [18].
ties of the laser field up to 1 kW/cm2 [12]. These results are in good agreement with the theoretical curve (Fig. 5) plotted within the model [16] for the Rabi frequency ≈1.3 MHz. With a further increase in the power PL, the Faraday rotation signal in exact resonance in the case L = λ almost vanishes, whereas the signal in the case L = λ/2 increases (dependence of the signal on PL is shown in the inset of Fig. 4), but its spectrum is broadened. Such a behavior of the signal at L = λ (dependence of φ on PL) can be characterized as a new feature of nonlinear Faraday rotation [1].
Figure 8 shows the spectrum of the Faraday rotation signal obtained at the reduction of the thickness L from 280 to (80 ± 5) nm. A decrease in the amplitude of the signal occurs more rapidly than follows from Eq. (3) because a decrease in L results in an increase in the number of collisions of atoms with the walls of the nanocell, suppressing Faraday rotation. However, it is
Fig. 5. (Color online) Theoretical spectra of the Faraday rotation signal at the same parameters as in Fig. 4. The Rabi frequency is Ω = 1.3 MHz. Strong optical pumping effect is observed at L = λ.
Fig. 6. (Color online) Spectrum of the Faraday rotation signal at L = λ/2, PL = 10 μW, and magnetic field B = 1.1, 2.5, and 5 G. The lower line is reference (indicates the frequency position of the 4–3' transition). JETP LETTERS
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Fig. 7. (Color online) Temperature dependence of the spectrum of the Faraday rotation signal in the range of 119–227°C at L = λ/2 and B = 5 G. At a small increase in the amplitude of the signal, its spectrum is strongly broadened (from 35 to 200 MHz). The lower line is reference. The inset shows the temperature dependence of the spectral width of the Faraday rotation signal.
more important that, at a decrease in L, the peak in the spectrum of the Faraday rotation signal is shifted to the red region (marked by arrows in Fig. 8) and an asymmetric low-frequency wing appears (as compared to the high-frequency wing) owing to the van der Waals interaction of Cs atoms with the windows of the nanocell. A red frequency shift was previously detected in selective reflection [19], as well as in the absorption and fluorescence spectra of Cs atoms in a nanocell [20, 21]. The Faraday effect is preferable to selective reflection in application to the investigation of the van der Waals interaction because the length of the vapor column involved in selective reflection is fixed and equals λ/2π ≈ 140 nm. As compared to the detection of the fluorescence spectrum, advantages are a narrower spectrum of the Faraday rotation signal and a pronounced peak, which allows a more accurate measurement of the redshift. Using the shift by ≈35 MHz of the peak of the Faraday rotation signal at L = (80 ± 5) nm and plotting a diagram similar to that presented in Fig. 1а in [21], we obtain C3 = (1.1 ± 0.1) kHz μm3 for the coefficient of the van der Waals interaction of an atom with the sapphire window of the nanocell (for the 6S1/2–6P1/2 transition). This value is in good agreement with the theoretical value from [18], but is a factor of 1.7 smaller than the value obtained by the fluorescence method [21]. We emphasize that a thickness of 80 nm is the smallest at which the Faraday rotation effect has ever been detected. In this case, the signal is detected in magnetic fields of several gauss, which can be of interest for nanoinstrumentation. To summarize, we have demonstrated that the Faraday rotation effect with the use of a Cs nanocell is JETP LETTERS
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Fig. 8. (Color online) Spectrum of the Faraday rotation signal obtained at a decrease in L from 280 nm to (80 ± 5) nm and at PL = 0.5 mW and B = 5 G. With a decrease in L, the peak of the spectrum is shifted to the “red” region (shifts are shown by vertical arrows) and an asymmetric low-frequency wing appears (as compared to the high-frequency wing) owing to the van der Waals interaction of Cs atoms with the windows of the nanocell.
a new useful tool for laser spectroscopy. The presented features of the Faraday rotation signal will also be observed in nanocells filled with vapor of other alkali metals (Rb, K, Na, etc.) We are grateful to A.S. Sarkisyan for fabrication of nanocells, as well as to A. Papoyan, A. Weis, M. Ducloy, and D. Bloch for stimulating discussions. This work was performed within the framework of the International Associated Laboratory IRMAS (Centre national de la recherche scientifique (CNRS, France) and State Committee for Science, Ministry of Education and Science of the Republic of Armenia). REFERENCES 1. D. Budker, W. Gawlik, D. Kimball, S. R. Rochester, V. V. Yaschuk, and A. Weis, Rev. Mod. Phys. 74, 1153 (2002). 2. V. V. Yashchuk, D. Budker, W. Gawlik, D. F. Kimball, Yu. P. Malakyan, and S. M. Rochester, Phys. Rev. Lett. 90, 253001 (2003). 3. B. Zambon and G. Neinhuis, Opt. Commun. 143, 308 (1997). 4. E. B. Aleksandrov, Phys. Usp. 53, 487 (2010). 5. J. A. Zielińska, F. A. Beduini, N. Godbout, and M. W. Mitchell, Opt. Lett. 37, 524 (2012). 6. M. A. Zentile, D. J. Whiting, J. Keaveney, Ch. S. Adams, and I. G. Hughes, Opt. Lett. 40, 2000 (2015). 7. M. A. Zentile, R. Andrews, L. Weller, S. Knappe, Ch. S. Adams, and I. G. Hughes, J. Phys. B: At. Mol. Opt. Phys. 47, 075005 (2014). 8. A. Sargsyan, A. Tonoyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, and D. Sarkisyan, Europhys. Lett. 110, 23001 (2015).
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Translated by R. Tyapaev
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