5.3 Feedback Bandwidth
5.3.3 Frequency Shift Measurement
The luxury of investigating the PZT mounting block outside the laser cavity is often not provided. In such cases, the frequency response of the PZT mirror can be determined without moving it out of the cavity by measuring changes induced in the laser’s frequency instead. The setup described here is designed for use with a mode-locked oscillator, but a similar measurement could be done with a CW laser. First, the pulses from the mode-locked laser are measured with a photodiode and band-pass filtered around the repetition rate frequency. For simplicity, assume the output is described by E1(t) =Aeiωrept+c.c., with the repetition rate ωrep ≈2π·1 GHz. The signal E1(t) is split into two parts, with roughly equal amplitudes (Figure 5.7). One signal is sent through a long cable, which leads to a temporal delay ∆t in this path. Combining both signals again with an electronic mixer gives
E2(t) = E1(t)·E1(t+ ∆t) = A2eiωrep∆t+c.c.+A2ei(2ωrept+ωrep∆t)+c.c.
| {z }
−→LP 0
,
≈A2cos(φ) = ˜E2(φ) with φ=ωrep∆t.
(5.22)
5.3. Feedback Bandwidth
Figure 5.7:The pulsed laser signal atωrep gets split into two parts. After the temporal delay of one signal, an electronic mixer superimposes both signals again. An Oscilloscope displays the modulation signal of the PZT and the low-pass (LP) filtered signal after the mixer for further analysis..
An analog low-pass filter removes the high frequency component at 2ωrep, leaving only the low frequency component of the signal. The cable length is chosen to correspond approximately to a phaseφn= (2n−1)π2 withn∈N, thus the Taylor expansion of ˜E2(φ) around φn yields
E˜2(φ)≈E˜2(φ)|φ=φn+∂E˜∂φ2(φ)|φ=φn(φ−φn),
=A2(−1)n(ωrep∆t−(2n−1)π2).
(5.23) Changing the cavity length with the PZT affects the round trip time of the pulse and consequently the repetition rate ωrep. Thus, a modulation of the PZT with the voltage U(t) = ∆U ei(ωmodt+φ01)+c.c. leads simultaneously to a modulation of ωrep. The frequency modulation is approximately ωrep = ωrep0 + ∆ωrepeiωmodt+c.c. with the mean repetition frequency ωrep0 = (2n−1)π2/∆t. Inserting the new repetition rate into Equation (5.22) yields
E2(t)≈A2(−1)n∆ωrep∆tei(ωmodt+φ02)+c.c.. (5.24) The digital demodulation of the signal with eiωmodt is analog to the previous description in Section 5.3.2, leading to the phase difference between the signal modulating the PZT and the response of the repetition frequency of the mode-locked laser. The response curve of the OFC used in this thesis is discussed later (Section 9.3.2).
Chapter 6
Saturated Absorption Spectroscopy
The short term stability of the precision laser system in this thesis is provided by the iodine standard, previously discussed in Section 5.2.2. This chapter explains the transfer of stability from a hyperfine-resolved iodine transition onto the reference laser by intro-ducing first the concept of saturation as the basis for Doppler-free spectroscopy. Laser modulation is the next step, to make use of the Doppler-free absorption signal and gener-ate a control signal for a feedback loop resulting in the laser stabilization (Section 5.3.1).
The laser system is ultimately stabilized by modulation transfer spectroscopy (MTS), which is gradually introduced in this chapter by explaining fundamental laser modulation techniques.
6.1 Einstein’s Rate Equations
Einstein’s rate equations provide general introduction into absorption, stimulated emission and spontaneous emission of discrete light photons of energy hν. Each interaction is associated with an Einstein coefficient. Spontaneous emission has the coefficient±A12, and stimulated processes are labeled with ±B12 where the indices correspond to the affected states of the transition. The sign represents the loss or the gain of atoms populating a particular state. A simple two-level atom is described by the rate equations[142]
dN2
dt =−A21N2−B21ρ(ω)N2+B12ρ(ω)N1, (6.1) dN1
dt =A21N2+B21ρ(ω)N2 −B12ρ(ω)N1, (6.2) where ρ(ω) is the field energy density. The total density number N =N1+N2 is constant, with N1 associated with the ground state and N2 with the excited state. Not-degenerate energy levels are simplified withB12 =B21, meaning the rates for emission and absorption are equal. Setting the change of the populations to zerodN1/dt= 0 anddN2/dt= 0 leads
6.1. Einstein’s Rate Equations to the steady state solution
N2
N1 = 1
A21/(B21ρ) + 1. (6.3)
At the limit of a strong electromagnetic fieldρ(ω)→ ∞the populations are equalN1 =N2 and no population inversion is possible. Therefore, in a two-level system, no laser operation takes place. In absorption spectroscopy, this is also the basis of saturation. The opposite limit of a weak field leads to a proportional relation between N2/N1 and ρ(ω). A more general description of saturation requires considering degenerate energy levels to derive relations between the Einstein coefficients. The number of the degeneracy of a specific state is described by the factorsg1 andg2. Additionally, the atoms are non-interacting and are in thermal equilibrium. Therefore, the average distribution is given by Maxwell-Boltzmann statistics, which leads to the general steady-state solution of[142]
N2
Solving this equation for ρ(ω) is essentially Planck’s distribution of black body radia-tion[143] These relations will be used soon in one of Einstein’s rate equations. First, it is convenient to redefine the energy density ρ(ω) = s(ω)ρ = s(ω)I/c with the line shape function s(ω). In case of spontaneous emission s(ω) is a Lorentzian function with the natural linewidth ∆ω =A21 describing the probability of spontaneous emission around resonance ω0. Inserting Equation (6.6) and the line shape function into Equation (6.1) yields[142]
dN2
where σ(ω) =A21λ2s(ω)/4 denotes the absorption cross section. The resonance condition ω = ω0 leads to the cross section σ = λ20/(2π). Therefore, the dimension is an area and Iσ(ω) describes the absorbed power by a single atom. Introducing the absorption coefficient[68,144]
Chapter 6. Saturated Absorption Spectroscopy
where ∆N denotes the population density difference, and this leads to Beer’s law
I(z) = I0e−α(ω)z. (6.9)
This equation describes the exponential decay of the initial intensity of a light field after propagating a distance z through an absorbative medium (Figure 6.1a). However, α(ω) is not independent of the intensity, which can be seen by calculating the steady-state solution of the rate equations again. For simplification, it is assumed the degeneracies are g1 =g2, and the field is on resonance. Thus, the absorption coefficient is[142]
α(ω0) =σ(ω0) N 1 + II
sat
, with Isat = ~ω0A21
2σ(ω0). (6.10)
At small intensities the absorption coefficient is nearly constant with σ(ω0)N, while at high intensities the absorption coefficient converges to zero. The saturation intensity Isat
corresponds to the intensity at whichα(ω0) decreases by a factor of two. Saturation effects are a double-edged sword in spectroscopy. Increasing the laser field intensity does not increase the absorption and the associated signal strength linearly. At some intensity the line profile gets distorted and the signal to noise ratio (SNR) decreases. On the other hand, these effects open the field of Doppler-free saturation spectroscopy.