Nonlinear magneto-optical rotation in optically thick media

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4the optimum initial saturation parameter κ 0 ≈ 1.8α 0 l,we have from Eq. (14) that κ(x) ≫ 1 for all x. For theoptimum light intensity, then,√(δB) −1 ≈ 0.8 gµω Atα 0 l¯hc≈ 1.1 gµ¯hγ√Alntγ . (18)This result is consistent with the general observation [3]that optimized NMOR provides a method for measuringa spin-system at the standard quantum limit (SQL) givenby spin-projection noise [23]. The sensitivity is then expectedto scale as the square root of the product of thenumber of available atoms and the spin-relaxation time(see, e.g., Ref. [18], Sec. 3.1.1), which is indeed the result(18).IV.THE DOPPLER-BROADENED CASENow we consider the case where the Doppler width Γis much greater than the power-broadened line width. Inthis case, we need to average over the over the atomic velocitydistribution, which is equivalent to averaging overthe Doppler-free spectral profiles. On resonance with theDoppler broadened transition, the absorption coefficientis given byα DB ≈ 1 √ πΓ∫ ∞−∞≈ α ∫ ∞0√ πΓ=√3π8γ 0Γ−∞α(∆)d∆d∆(2∆/γ 0 ) 2 + 2κ/3α 0√ κ. (19)In this case the Doppler-broadened unsaturated absorbtioncoefficient is given in terms of the Doppler-free unsaturatedabsorbtion coefficient by√ πα 0 | DB=2γ 0Γ α 0. (20)Comparing Eqs. (19) and (7), we see that we have reproduceda well-known result (see, e.g., Ref. [19], Sec.7.2.1) that resonant absorption falls as 1/κ for Dopplerfreemedia and as 1/ √ κ for Doppler-broadened mediawhen κ ≫ 1. The change in κ per unit length isdκ(x)dx= −α DBκ(x)√6π γ 0≈ −4 Γ α √0 κ(x), (21)and solving for κ as a function of position,( √ ) 2√ 6π γ 0κ(x) ≈ κ0 −8 Γ α 0x . (22)Note that the behavior of the saturation parameter as afunction of distance is different in the Doppler-broadenedcase compared to the Doppler-free case [cf. Eq. (14)]where the saturation parameter falls approximately linearlywith distance.Taking the average of small-field rotation over theDoppler distribution givesdϕdΩdx∣ ≈ 1 ∫ ∞√DBπΓ −∞≈ α ∫ ∞0√ πγΓ≈√6π4γ 0Γ−∞dϕdΩdx (∆)d∆1(2∆/γ 0 ) 2 + 2κ/3 d∆α 0γ √ κ . (23)We see that the rotation per unit length scales as 1/ √ κ,in contrast to the 1/κ scaling for the Doppler-free case[similar to the situation with absorption, Eqs. (7,19)].This is because the number of atoms producing the effectis, in a sense, not fixed; with increasing light power, alarger fraction of the Doppler distribution is involved.Substituting (22) into Eq. (23) to find the contributionto the slope as a function of position givesdϕdΩdx (x) ∣ ∣∣∣DB≈ 2 γα 08 √6πΓγ 0√κ0 − α 0 x . (24)It is interesting to note that while the light power andthe rotation slope per unit length behave differently inthe Doppler-broadened case compared to the Dopplerfreecase, Eq. (24) has the same functional form as forthe Doppler-free case [Eq. (15)].Integrating over the length of the medium, we obtain∫dϕl∣ dϕ ∣∣∣DBdΩ∣ ≈DBdΩdx (x) dx≈ 2 γ0∫ l0√8Γ6π γα 0√κ0 − α 0 x dx(= 2 √γ ln κ0√κ0 − √ 6π γ 0Γ8α 0 l). (25)The behavior of (25) as a function of α 0 l is qualitativelysimilar to that of Eq. (16) shown in Fig. 2. However, thedependence on κ 0 is different.The magnetometric sensitivity is given by(δB) −1DB= gµω¯hc≈ gµω¯hc= gµω¯hc√dϕ2AtdΩ∣ DBπ γκ(l)√8Atπγ√ √32( √ ) ( √ )√ 6π γ 0κ0 −8 Γ α κ00l ln √κ0 − √ 6π γ 0Γ8α 0 l( )At γ 0pγ Γ α 0l(p − 1) ln , (26)p − 1
5with the change of variablesThe factorκ 0 = 3π (p γ )0 2.64 Γ α 0l(27)( ) p(p − 1) lnp − 1goes to unity as p goes to infinity; it is approximately 0.9at p = 5. (Further gain from increased power is minimal,and if the power becomes too high, the approximationof the Doppler width being much larger than the powerbroadenedwidth breaks down.) Thus, in this case,√(δB) −1DB ≈ 0.8gµω ¯hc≈ 0.3 gµ¯h λlnγ 0ΓAt γ 0γ Γ α 0l√Atγ , (28)for sufficiently high κ 0 .This result, where sensitivity increases linearly withoptical thickness, holds for the case where the powerbroadenedwidth ∼ γ 0√κ(x) is smaller than the Dopplerwidth for all x within the sample, i.e.1 ≪ κ 0 ≪ (Γ/γ 0 ) 2 andΓ/γ 0 ≪ α 0 l ≪ (Γ/γ 0 ) 2 , (29)since α 0 l is related to κ 0 by Eq. (27). As power andoptical thickness are increased beyond this range, i.e.κ 0 ≫ (Γ/γ 0 ) 2 andα 0 l ≫ (Γ/γ 0 ) 2 , (30)we obtain the Doppler-free case, where sensitivity increasesas the square root of the thickness [Eq. (18)].V. THE GENERAL CASEA numerical result can be obtained for the general casewhere the restrictions (29,30) on κ and α 0 l are removed.In a typical experiment, light power is ∼ 1 mW, the laserbeam diameter is ∼ 0.1 cm, λ ≈ 800 nm, and Γ/γ 0 ≈ 60.Thus the effective ground state relaxation rate due to thetransit of atoms through the laser beam is γ ≈ 2π·50 kHzand the initial saturation parameter is κ 0 ≈ 4 × 10 3 .Here, as in a typical experimental procedure, the opticaldepth is varied (by changing atomic density) whilethe laser power is kept constant. Normalized transmission,differential small-field rotation, total small-field rotation,and magnetometric sensitivity are plotted in Fig.3 as a function of optical depth. For small optical depth,κ > (Γ/γ 0 ) 2 and the medium is effectively Doppler-free.Transmission (∝ κ) falls linearly until the transition tothe Doppler-broadened case is made (dashed line). ThendφddΑ0{Γ Κ0310 3a210 3110 3310 30b210 3110 306c42040d2000 210 3 410 3 Α610 3{810 3 1010 3410 3dφdΓ Κ dφdΓFIG. 3: Normalized (a) saturation parameter, (b) differentialsmall-field rotation, (c) total small-field rotation, and (d) inversemagnetometric sensitivity as a function of optical depthwith initial saturation parameter κ 0 = 4×10 3 and Γ/γ 0 = 60.Plots (b), (c), and (d) are in units of radians. The dashed lineindicates the transition from the Doppler-free to the Dopplerbroadenedregime [κ = (Γ/γ 0) 2 ], and the dotted line indicatesthe point at which non-linear effects begin to turn off (κ = 1).Linear optical rotation is neglected in this plot. The solid lineindicates the optical depth at which maximum sensitivity isachieved.transmission falls quadratically until the linear regime isreached (dotted line), after which it falls exponentially.Differential small-field rotation (∝ dϕ/[dΩd(α 0 l)]γ) initiallyrises, as κ falls and power broadening is reduced,until non-linear effects begin to turn off. (Linear opticalrotation is neglected in this plot.) Since magnetometricsensitivity depends both on total optical rotation andtransmission, an intermediate value for the optical depthproduces the greatest sensitivity (solid line). Multiplyingthe normalized inverse sensitivity dϕ/dΩ · γ √ κ by2 √ 2π gµ¯h√A(λ √ γ ≈ 108 G/ √ ) −1Hz (31)
Page 1 and 2: Nonlinear magneto-optical rotation
Page 3: 3The sum of the two contributions p
Page 7: 7series in pure and applied optics

Nonlinear magneto-optical rotation in optically thick media

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