A home-built lock-in amplifier for laser frequency stabilization

912 Can. J. Phys. Vol. 83, 2005Fig. 4. (a) Relative sizes of the narrow resonance L (broken line) given by (2), the Doppler background G(thin line) given by (6) and the saturated absorption signal S 2 (bold line) modeled as S 2 = L + G. (b) Firstderivatives of L, G, and S 2 .(c) Third derivatives: Compared to the other functions, the third derivative of Gis so small that it does not appear on the graph. Consequently, the third derivatives of L and S 2 overlap.a)b)c)signals (arb. units)0ω oωsignals (arb. units)0 ωω osignals (arb. units)0ω oωThe constants γ and λ are shape parameters that control the amplitude and width of the Gaussian,respectively. The saturated absorption signal S 2 can, therefore, be modeled asS 2 (ω L ) = L (ω L ) − G (ω L ) (7)If the first derivative of the saturated absorption signal is used as feedback, the background componentof the signal would cause the zero crossing to be shifted to a frequency different from ω 0 . However,since the background is approximately linear in the vicinity of the resonance, it can be greatly reducedby taking the third derivative of the signal, as illustrated in Fig. 4. The third derivative of a resonancepeak resembles the first derivative and, therefore, is also suitable for locking.We now show how the third-derivative error signal is obtained by mixing the absorption with thethird harmonic of the modulation and filtering out the AC components. The signal given by (7) is writtenas a third-order Taylor expansion about ω c , shown below.S 2 (ω L ) ≃ [G (ω c ) − L (ω c )] + A cos(t) [G (ω c ) − L (ω c )] ′ + 1 2 A2 cos 2 (t) [G (ω c ) − L (ω c )] ′′+ 1 6 A3 cos 3 (t) [G (ω c ) − L (ω c )] ′′′ (8)The mixer output is modeled as the product S 2 (ω L ) [B cos(3t + φ)], where B and φ are theamplitude and phase of the third-harmonic modulation signal, respectively. The constant terms ofthe expanded mixer output become the error signal E 3 (ω L ). Again, using the trigonometric identitycos(x) cos(y) =2 1 [cos(x + y) + cos(x − y)], it is clear that the error signal will be proportional onlyto terms in S 2 with the factor cos(3t). S 2 contains a sum of terms containing nth derivatives of thesaturated absorption signal multiplied by cos n (t). However, cos n (t) can be expanded, using knowntrigonometric identities, as a linear function of cos terms with angular frequencies n, n − 2, ... ≥ 0.Consequently, it can be shown that the error signal E 3 (ω L ), shown below, is proportional to the thirdderivative of the absorption signal at ω c .E 3 (ω L ) ≃ 148 A3 B cos(φ) [G (ω c ) − L (ω c )] ′′′ (9)As discussed, both the first- and third-derivative error signals are suitable for locking the laserfrequency to the peak of an atomic resonance. Although the third-derivative signal has a lower signalto-noiseratio, it automatically reduces the effect of a Doppler background. Additionally, since the widthof the signal close to resonance and between the turning points is narrower than for the case of the firstderivative it can be used to obtain a more precise lock. However, this results in a smaller recapture rangefor correcting the laser frequency.© 2005 NRC Canada