Modulation Transfer Spectroscopy

The previous sections aimed to give a basic understanding of fundamental modulation techniques, carrying the message that high modulation frequencies reduce the amplitude noise^[157,158] which results in a higher SNR.

However, independent of the modulation frequency a complete cancellation of the amplitude background noise is not possible by design. A laser is a very coherent light source, thus placing optics inside the path of a laser beam always results in spurious interferometric fringes. The linear absorption of these unwanted parts of the electromagnetic field adds up to a noisy background. The solution for this issue is to make the detection schema insensitive to linear absorption features. Modulation transfer spectroscopy (MTS) can cancel the background and increases the sensitivity close to the shot-noise limit^[158]. The setup of MTS (Figure 6.4) looks at first sight very similar to FM spectroscopy (Figure 6.3) with the obvious addition of a counter propagating beam. However, the mechanism behind

Figure 6.4: Schema of a MTS setup. The saturation beam coming from the left interacts nonlinearly with a counterpropagating probe beam inside the absorbing sample. The modulation of the saturating beam is transferred onto the probe beam for phase sensitive detection.

it is described by a nonlinear process of the third order, called four-wave mixing^[157,159]

(Section 4.0.4). The modulated laser beam saturates the transition. It contains the carrier

Chapter 6. Saturated Absorption Spectroscopy

frequency and two sidebands like in the case of FM spectroscopy (Equation 6.19). The counterpropagating probe beam is also at the carrier frequency but is unmodulated and at lower power. The saturation of the transition causes the nonlinear power dependence of both beams, which results in a nonlinear interaction and a transfer of the modulation frequency of the pump beam onto the probe beam. Modulation transfer happens only at resonance. Therefore, the background noise becomes independent of linear absorption effects and leads to a zero baseline. The mathematical description makes use of third order perturbation theory, which leads to the signal^[160–163]

S(ω_m) = c with the Bessel functionsJ_n of ordern, the phase modulation indexM and the Lorentzian resonance functions

L_n= Γ²

Γ²+ (∆−nM)² and D_n= Γ (∆−nM)

Γ²+ (∆−nM)². (6.24) These functions describe the absorption for a detuning ∆ away from the resonance for a natural linewidth δ and a modulation frequency ω_m. The detection of the signal is dependent on the phase of the demodulation signalφ which is one of the possible tuning parameters to maximize the signal^[163]. Demodulating the signal with cos(ω_mt+ϕ) leads again to a in-phase and a quadrature part of the signal (Section 6.3.2). The maximum signal is found at ϕ > 0, which requires a linear combination of both parts. However, sometimes it is convenient to generate either the pure in-phase or quadrature signal.

Maximizing the signal is not necessarily the target. In case of laser locking, one might prefer a signal with a large slope at zero crossing over a signal with a larger amplitude.

Considering a modulation index M < 1 simplifies the evaluation of the signal because only the first order sidebands contribute^[164]. For a small modulation ω_m/Γ ≈ 0.5 the quadrature and in-phase signals have the same line shape (Figure 6.5a and 6.5b). The in-phase component consists of Lorenzian functions Ln describing the absorption and the quadrature component describes the dispersion with D_n. This characteristic is analog to the case of FM spectroscopy in Equation (6.22). Therefore, the in-phase component is proportional to the first derivative of the absorption, and the quadrature component is proportional to the second derivative of the dispersion^[153]. The maximum gradient is reached at ω_m/Γ = 0.35 and at ω_m/Γ = 0.67 for the in-phase and the quadrature compo-nent, respectively. A further increase of the modulation frequencyω_m raises only the peak to peak amplitude, and the slope starts to decrease after passing a modulation frequency of ω/Γ≈1.4. Much larger MTS signals are achieved by increasing the modulation index

6.3. Laser Modulation Techniques

(a) (b)

Figure 6.5:Calculation of the MTS (a) in phase and (b) quadrature signal after demodulating with cos(ωmt+ϕ).

The modulation index is set toM= 0.5.

M >1. Essentially, the behavior of the in-phase and quadrature component are the same, as in the case of M < 1. The difference is a faster decline of the signal amplitude with respect to the modulation frequencyω_m (Figure 6.6a-6.6b). Choosing an ideal signal based

(a) (b)

Figure 6.6:Calculation of the MTS (a) in phase and (b) quadrature signal after demodulating with cos(ωmt+ϕ).

The modulation index is set toM= 3.

on this theoretical curves, one should select the in-phase signal. The modulation index should be M >1 and the modulation frequency wm smaller than the natural linewidth Γ.

Chapter 7

Spectroscopy on a Molecular Beam

The molecules are prepared in a highly-collimated molecular beam of the hydroxyl radi-cals (OH), which is crossed perpendicular by the spectroscopy laser. Afterwards, electron-ically excited OH molecules emit fluorescence light on a microsecond timescale, which is detected with a photomultiplier tube (PMT). Unfortunately, the excitation frequency of the molecules is shifted, due to the Doppler-effect (Section 7.2). The same effect dominates the line broadening of the transition frequencies. For instance, the natural linewidths of the A²Σ⁺,v⁰ = 0←X²Π_3/2,v⁰⁰ = 0 transitions in hydroxyl radical (OH) are

∆ν = 1/(2πτ)≈231 kHz^[165], based on a lifetime ofτ ≈688 ns^[166]. Saturation broadening increases the linewidth to ∆ν⁰ = ∆ν√

1 +S₀ ≈400 kHz for an overestimated saturation parameter S₀ = 2, which corresponds to twice the light intensity required for steady state absorption and emission^[123]. Now, in contrast, the full width at half maximum (FWHM) due to the Doppler effect is approximately 8 MHz. Individual transitions start to blend and make the determination of their center frequencies challenging. The Zeeman-effect (Section 2.6.1), as well as laser field dependent effects (Chapter 8) are within the line broadening. Therefore, this chapter explains the molecular beam and its relation to the first order Doppler-effect.